This is a guide mostly intended for players who are just starting out using Skill Inheritance who aren't sure which skills are useful and who to use them on.
I wrote this because the last guide on this I've seen is 4 months old and focuses more on covering every skill briefly, here I'll only mention skills that I think are very useful for at least a couple of units (and some that people often seem unsure about), but in a bit more detail.

At the end of this guide (to not clutter the main text) I'll list which 4 star units have each of the best of these skills. To see all the heroes who learn a certain skill, just search for it on the wiki, it'll show you who learns it and when they get the final rank.
Unless otherwise specified, I'm always referring to the max rank versions of each skill.

Feedback or suggestions of any kind are very welcome!

 

Weapons

What's inheritable: among 5* weapons (highest rank weapons, those you can only learn at 5 stars), basically every weapon that has a + at the end and every healer weapon is inheritable (but only to heroes who use the same kind of weapon; dragons can inherit all -breath weapons regardless of color), the ones that don't have it are called “legendary” weapons and can't be inherited. E.g. Slaying Edge+, Emerald Axe+, and even Clarisse's Bow+/Berkut's Lance+ are all inheritable, Siegfried, Falchion, Dark Excalibur etc. aren't.

What's good:

• Brave weapons of any kind are good on high Atk units to deal a ton of damage in player phase. Particularly Brave Bow+ on Bride Cordelia or Leon scores a lot of ORKOs (one round KOs). Most units want Death Blow with them to deal maximum damage, fast units may want Life and Death to make up for the speed loss and get the possibilty of 4-hitting slower units.
• -blade tomes (from Nino, Tharja, and Odin) are generally the best weapon for mages who don't have legendary tomes. They increase your Atk for any buff you have (those that are shown in your stats and are applied at the start of the turn, like Hone/Fortify buffs or Defiant X, not those that say they work “in combat”, like Spur X buffs. So e.g. Hone Spd 3 increases your Atk by 4, Hone Atk by 8 (the buff itself + the weapon bonus). Eirika or Ephraim can boost your Atk by 14 in total with their weapon, a Hone/Fortify buff and a Rally assist). Increase your special cooldown by one.
• -raven tomes give triangle advantage against colorless units, together with Triangle Adept you beat 2 of the 4 colors pretty hard, but lose out on the power of other weapons + a stat boosting A passive, so this isn't too useful for most units.
• -owl tomes: 3 lower base power than -blades but no special charge penalty, boost Atk/Spd/Def/Res by number of adjacent allies x 2 in combat. Lower power and harder to use because of the “in combat” part (you can't just run ahead of your other units and attack or won't get the bonus), but especially boosting speed can help for getting or avoiding doubles. Not entirely sure who these are best for. Sound good for defensive mages that want to get attacked first and counter, positioning for that is easier and the Res/Def boost helps, but those mages may prefer raven + TA. I guess it's good for mages that want to do both, with -blade being better on offensive units like Linde/Nino/Tharja and raven/TA on defensive ones like M!Robin, maybe owls are best on e.g. Mae (who has one naturally) or Soren.
• Gem weapons (Ruby Sword, Sapphire Lance, Emerald Axe) boost triangle advantage, good to counter one color strongly, but miss out on power against the rest. Useful on Azura and Olivia because they aren't expected to beat anything without triangle advantage anyway and this makes them way better at it, generally not so good on most other heroes.
• Firesweep Bow/Lance: prevents others and yourself from countering, good e.g. on Setsuna or Summer Gaius who're fast enough to double most units to deal a lot of damage without taking a counterattack.
• Wo Dao+ is a great sword for fast red units, unfortunately they are often outclassed by units with legendary weapons, like Ragnell/Raijinto or Falchion.
• Legion's Axe changes buffs to penalties on the opponent after battle. Sounds great, but you won't attack many units more than twice to kill them, so somewhat limited in practice in my opinion.
• Lightning Breath is the dragon weapon with distant counter, generally the best choice for most/all dragons (sometimes Dark Breath is better to debuff enemy units).

For the highest rank weapons you always need to sacrifice a 5* unit, so weapons are expensive to get, but generally have the biggest impact on a unit's strenght and how they play, so it's worth it for strong heroes that start out with a weapon they can't use too well.

Assists

What's inheritable: Everything except Dance/Sing; healing skills can only be inherited by healers, while they can't learn any other assists.

What's good: For non-dancers/healers, movement skills are usually the best, with some exceptions.

• Reposition is probably the best movement skill, moves a unit on the opposite side of you, great to bring them back to safety. Particularly good for attacker + dancer combos to get the dancer to safety while retreating and leaving the attacker in the front for good positioning for your next turn.
• Draw Back: basically the same as Reposition, but your units end up in switched positions. Equally good to get others out of danger, arguably worse in combination with a dancer.
• Swap: swaps places with ally. Good on defensive units, gets the ally to safety and sets the user up to get hit by an enemy in a favorable matchup. The standard choice for all armors. Reposition/Draw Back often might not work well depending on terrain/obstacles, Swap is very reliable.
• Ardent Sacrifice/Reciprocal Aid: Mainly used to get into Desperation range safely. I'm not a big fan of it, since a different unit needs to enter combat and take damage to make using it possible, and you sacrifice another turn to set it up, so it seems hard to do consistently without sacrificing kills or good damage. Still worth mentioning here though.
• Rally skills: useful for buffing blade tome mages because you get twice the benefit out of it, otherwise not too impressive. “Rally Attack Speed” and “Rally Defense Resistance” are notable because they buff blade tomes more than the others and they are currently the only stage 2 assists and believed to give more points in arena than all other assists (probably 1 point more, on average).

Out of the healing skills, Rehabilitate is usually considered the best, mainly for its ability to get very low health units back to full HP.

Moonbow Specials

For the exact math of which special will deal how much damage you can go e.g. to this post.
General thoughts: Defensive specials are usually not recommended, since they are specific to either ranged or melee attacks, and even when that works out, by the time they can activate you might just be dead to the lowered damage anyway. An exception may be using them together with Shield Pulse, I discuss that in the Passives section. As for offensive skills:

• Moonbow is the only reliable one (i.e. not Reprisal) with a cooldown of 2, and works well on basically every unit. The low cooldown means you'll probably activate it on every map where the unit attacks. It lowers the opponent's Def/Res instead of boosting your Atk by a set amount based on your own stats. This means that it'll be strongest against units with a high defensive stat, which is where you generally need a strong special the most. It's also great for units with relatively balanced stats or brave weapon users without a strong stat except Atk, because you don't need a strong stat to make this skill work well. Used with the Quickened Pulse sacred seal it always triggers the first time a brave weapon user attacks on a map. This can increase their immediate kill potential by a lot, there's e.g. almost nothing that survives a Reinhardt or Brave Bow+ B!Cordelia/Leon initiating with this setup.
• The notable 3 cooldown specials are Draconic Aura (deals extra damage based on your Atk stat), Bonfire (same with Def), Iceberg (Res), and Luna, which is a stronger version of Moonbow. If your hero has outstanding Atk/Def/Res you can pick one of these accordingly for significantly more damage against most units compared to Moonbow, but be aware that they'll trigger less often than Moonbow, sometimes not at all.
• Dragon Fang/Ignis/Glacies are the stronger 4 cooldown equivalents of Draconic Aura/Bonfire/Iceberg, but the extra cooldown can be a big drawback, while the extra damage is often just overkill anyway. Can be good on bulky slow units, particularly with Quick Riposte to enter the second fight with a strong special (although a 3 cooldown special guarantees you kill basically anything that doubles (but doesn't kill you) in the first fight), on offensive units I wouldn't use them.
• Damage-dealing 5 cooldown specials tend to be bad because when you get to use them, you've usually basically won the map anyway. I wouldn't use them.
• Galeforce (5 cooldown, if it activates you can move again that turn) can work on fast brave weapon users and you can do ridiculous stuff like this with it, but it seems overall pretty situational.

Passives

A Slot

Offense oriented:

• Fury: grants +3 Atk/Spd/Def/Res, you take 6 damage after combat. One of the best skills in the game for both offense and defense. 3 extra speed and attack help a lot offensively, the Spd boost also helps avoid a bunch of doubles on many units, and the Def/Res boost mostly balances out the HP loss in interactions where you take damage. Particularly good on squishy fast mages, they get them more damage, a bunch of important doubles, and boosts their weak defense enough to survive a counter hit from almost everything. The recoil damage can even be beneficial for getting into Desperation range safely, which is particularly useful for clearing Tempest Trials or chain challenges. Dancers also like Fury because it gives them killpower without really making them weaker defensively than with e.g. HP+5, considering they won't take many hits anyway. Defensive units like Nowi can benefit a lot from it too, both by boosting their defenses and getting an extra 3 Atk, for 6 extra damage in their first interaction if used together with Quick Riposte.
• Death Blow: +6 Atk when initiating, the standard choice for all Brave Weapon users (unless they want to keep using their high speed like Bride Cordelia, then Life and Death might be preferred, see below) to maximize their damage output and ORKOs in player phase, and possibly useful for some slow but strong attackers that try to score many OHKOs, like Lilina.
• Life and Death: grants +5 Atk/Spd, -5 Def/Res. A great offense boost, in exchange for lowering your defense significantly (although because of the Spd boost you'll still survive some matchups you wouldn't without it). Useful e.g. on units with decent Atk and Spd they need to boost hard to make them great, but defenses that are strong enough to take a hit or two even with LaD, like Minerva; arguably on units like Delthea and maybe -Def/-HP Linde who have low HP and one defensive stat (Def in this case) that's so low they'd die to many things even with Fury, but another one (Res here) that's strong enough to take a hit even with LaD, for that extra power over Fury; or on fast brave weapon users like B!Cordelia who want to keep using their speed to potentially quad hit enemies, and if they have at least one strong defensive stat, to not be as vulnerable to fast units in the enemy phase.
• Swift Sparrow: +4 Atk/Spd when initiating. Good for many who like Fury or Life and Death. Lacks the offense boosts in enemy phase though when countering, and Fury's defensive advantages. A bit weaker than LaD too, but without the drawbacks. Only available from Katarina, who's 5* only, and Spring Lucina, who was a limited unit, so a pretty rare and valuable skill.

Defense oriented:

• again, Fury. It's also a great skill on more defensive units, for reasoning see above.
• Triangle Adept: Gives yourself and opponents +/-20% extra Atk to weapon triangle advantage/disadvantage. Does not stack with Gem weapons or enemy unit's TA skills. Great to counter a color hard, with good defenses they'll deal very little or no damage at all to you, but it leaves you basically unable to do anything against the color you're weak to. You're also at a slight disadvantage against your own color and colorless units because you're missing the opportunity to use the slot for a stat boosting skill. Overall probably best for -raven tome users because they can beat two colors well with it (although against their own color the combination of a somewhat weak weapon and a useless A skill will make them rather weak), also good if the rest of your units are just pretty weak to a color and you want a reliable way to beat them, ideally on a unit that doesn't strongly rely on a different A skill normally (Sanaki, who won't get her speed to great levels regardless of skills and is part support unit because of her weapon anyway is a good example). Also useful on Nowi, with it she can beat Falchion users and beats other reds very strongly, although Fury is better against most non-reds. Other units who can use it well are Fae, Xander and Tiki (either one).
• Distant Counter: The generally best A slot skill for all armors except maybe brave lance Effie, also useful on other defensive units, although they are mostly outclassed by units who get weapons that naturally counters ranged enemies and still have a free A slot, so Xander/Camus, Ryoma/Ike and the dragons. Titania is a notable hero who can use it well, alone she's mediocre but in horse emblem she's pretty strong. Unfortunately only available on Hector.
• Close Counter: Only available on Takumi. Unlike Distant Counter, not that great of a skill in my opinion. Ranged heroes suffer a stat total penalty, so the ones who can take hits to make it usable lack the offense to really kill things with their counter hits (Quick Riposte helps a lot there though). Before skill inheritance it was pretty strong on Takumi, but now that everyone can use various skills to boost their offense and there are some good brave weapon users he dies a bit too easily to be able to use it as well as he used to. I can't even think of any high or top tier hero who could use it better though.
• Distant/Close Defense: receives +6 Def/Res during combat when attacked by ranged or melee units respectively. Distant Defense is only available from the 5* only Celica, Close Defense from Summer Tiki, who's also limited. Both seem pretty good on defensive units, but do nothing when you initiate yourself or when the wrong type of hero attacks you, and don't provide any offensive benefits at all. Fury is often better because it's more universal and helps when attacking/countering too, but if you want to build a unit very defensively these skills can be good as well. Examples are Fae and Xander who can use Distant Defense instead of Fury or Triangle adept to tank archers and mages of their own color really well, Xander in particular doesn't miss the loss of Fury's Def too much against swords and axes because of his already great defense stat.
• Iote's Shield: removes weakness to bows on fliers. Obviously good if you're facing fliers, but you're missing out on a helpful A skill against everything else. With some brave weapon fliers you can just use Death Blow or Life and Death and ORKO archers anyway, you just can't let them initiate on you, which you usually don't want either way. I wouldn't use this skill on anyone you're heavily investing in, unless you're building flier emblem and need a very reliable way to beat archers. Grani's/Svalinn Shield are much worse because they share the same problem, but weapons effective against cavalry or armors are very rarely used.

B Slot

• Desperation: When at <=75% HP, potential follow-ups happen immediately after your attack (before the enemy counterattacks). The standard B skill for fast glass cannons, e.g. Linde/Nino/Tharja. Ardent Sacrifice/Reciprocal Aid can help set this up, but are awkward to use as said above. For arena or single maps, to use it you usually first attack someone you double and 2HKO who can counter, but not kill you, with Linde e.g. Brave Bow users, Ike, or slower red mages. Their counterattack and potentially Fury damage puts you below 75% and activates Desperation, with that active you can e.g. kill Kagero, who'd otherwise counter before your second hit and kill you easily even from full health. For Tempest Trials and chain challenges Fury is a great and reliable way to get you into range for it, which makes these units very useful for clearing them.

• Quick Riposte: When attacked at >=70% HP, unit counterattacks twice. Amazing skill for defensive units, with decent attack you can bait most units you aren't weak to and KO them with the counterattacks. Usually only works once against units of your/grey color because of the HP restriction, but e.g. TA Nowi can keep using it against multiple non-falchion reds. Most useful on bulky units with some form of distant counter because it will affect everyone attacking you. Usually the best skill for the dragons, and Ike not running Swordbreaker.

• -breaker skills: At >=50% HP against user of the respective weapon, unit makes a follow-up attack, the opponent does not. Inheritable unless you have triangle disadvantage against that weapon class (e.g. Hector can't inherit Swordbreaker or Red Tomebreaker, but all other -breaker skills). Using Swordbreaker on sword units themselves or other reds can improve their matchup spread a lot if they don't have great speed to get those follow-ups normally, or even get doubled by many of them. E.g. Eldigan can win against all swords with default skills except Zephiel. It's similar for Lancebreaker on blues and Axebreaker on greens, although those units tend to be a little less common. Archers (and dagger users and healers) can carry all -breaker skills, with a brave bow they get to quad hit every unit of that weapon class that don't have skills that cancel it out, so even just using rank 1 or 2 of those skills can enable you to beat almost an entire weapon class (e.g. Leon and Bride Cordelia with Death Blow can both ORKO even Hector with Axebreaker).

• Vantage: Lets you counterattack first when attacked at <=75% HP. Most useful on units with some form of Distant Counter, most notably fast ones like Ryoma who don't get much use out of Quick Riposte, or Hector who has QR2 built into his weapon. Improves a lot of 1v1 matchups by giving you an extra hit, since you can attack so that your enemy's counterattack puts you near death, and when they attack you next in the enemy phase you get another hit in with Vantage, potentially killing them before they kill you. May kill multiple (generally squishy or weakened, more if you have a special ready) units on a map that would otherwise kill you first. Since you have to enter combat at least once to get into the HP range for it though, enemy units may often have others of your units in range to attack instead, so it's not always quite as useful as this sounds. Still a very good skill, probably even more so in arena defense teams.

• Wary Fighter: only usable by armor units. Prevents follow-ups in combat from both yourself and enemies if unit's HP is >=50%. Great for survivability with little drawback because most armors are very slow, one of the standard skills (the other big one being Quick Riposte) for all of them except Draug (because he's fast) and Hector (because it would clash with his weapon's built in QR).

• Wings of Mercy: enables unit to teleport to allies with HP <=50%. Great on dancers because they don't really need a combat B passive and appreciate the option to dance units that would normally be out of their range. Also good on healers, but they have great healer-specific skills too. Not too useful on other units because you can't move after the teleport, so attacking can be tricky, and most prefer having a skill that improves combat over this one.

• Hit and Run/Drag Back: Melee weapon users only. After initiating attack, unit moves back one space. For Drag Back, the unit you attacked moves into the space you previously stood on. If you kill the unit you attack the skills are identical, and Drag Back is more easily available. Can help you get out of enemy range after killing a target, particularly good on fliers because depending on terrain this is easier to do. No effect in combat though, so other skills can often be better, but often still a good choice on fliers particularly if there's no other skill for this slot you really need.

• Cancel Affinity: not inheritable for tome and staff users. Negates effects of unit's Triangle Adept or Gem weapon, so don't use them together. If affinity disadvantage exists, the weapon triangle affinity granted by foe's skills is reversed. This means that e.g. as Lucina against Azura with her Sapphire Lance, regular triangle disadvantage and the reversed effect from her weapon cancel each other out, same against blues with Triangle Adept 3. Enables you to have even matchups with Gem weapon or TA users that would normally strongly counter you, but there aren't a lot of those (notable are Azura and many Nowi and M!Robin sets for reds, Titania and Fae's best sets for blues, Sanaki, Ruby Sword Olivia and some Tikis for greens, and some M!Robin and Cecilias for colorless). I probably wouldn't use this skill over more versatile ones, but if you have really big problems with any of these units and there's no must-have B skill for your unit, I guess it could work okay.

• Seal Attack/Seal Attack Speed: lowers opponent's Atk by 7 / lowers their Atk and Spd by 5 after battle. Reduces the opponent's offense considerably for the next turn, but is still far from negating it, so I think there are better options.

• Shield Pulse: if your special triggers based on a foe's attack, special cooldown is reduced by 2 at the start of turn 1, unit takes 5 less damage when special triggers. Together with Aegis, improves your turn 1 survivability against most notably Reinhardt and Brave Bow users, or faster mages by a lot (Hector can take on any Reinhardt set even with a +6 Atk buff). This seems like a pretty good option to counter those units specifically, but is generally weaker than other special + passive combinations after the first interaction, and doesn't do anything against melee attackers. To me it sounds possibly worth using if you really have no other good way of dealing with those units, but are well set up against any other teams. I wouldn't use it with a special that reduces damage from melee attacks, because it's usually easier to work around those.

• Healer skills: Live to Serve heals the user by the amount you heal (good for staying alive when the healer needs to take hits too). Wrathful Staff makes damage calculation work the same way as with non-healers (so you do twice the damage as without it). Dazzling Staff (only available from Bride Lyn) prevents units from countering you, since you can't combine this with Wrathful Staff it's not great for just dealing damage, but in combination with B!Lyn's weapon it can be used to safely let the rest of your team kill a unit. All three of these skills can be good, depending on the healer you're using and the rest of your team.

C Slot

I'm only making some short comments here because what's best here mostly depends on your team:

• If running a horse/armor/flier only team, obviously use the buffs specific to them. Even with just running two units of that kind, they can be well worth it.
• Hone Atk is great, I'd use it on pretty much any team that doesn't run movement-restricted buffs.
• Run other buff/debuff auras as needed, particularly Hone Spd is often useful for offense on fast, and for defense on medium speed units; Spur Atk on top of Hone Atk can also be pretty good, but is slightly harder to use well; Threaten Def or Res can be good on more defensive teams to score easier kills, but since it'll often only apply after the enemy units attacked once it's harder to use on offense oriented teams. You can bait melee units to move within 2 spaces of you without getting attacked, but for cavalry or units with movement assists that can be hard, especially with obstacles, for armors it's easy though.
• Drive Atk is good and easy to use, but only available from Delthea, who you generally don't want to sacrifice for it.
• Panic Ploy is great at least on paper, especially against horse teams. In practice it's not always that simple to use if you only have it on one unit, if you have it on multiple ones you're missing out on crucial buffs. Sadly only available from the 5* only Luke.
• Resistance Ploy seems good on high Res mages like Linde, but doesn't do anything against very high Res units where you'd need it the most.
• Defense Ploy seems good on dagger users.
• Savage Blow seems like it could help a lot for softening up targets for later OHKOs, and break the 100% HP conditions of the weapons from Celica and the Ylissean Summer banner units. I don't think I have enough experience with this skill to judge exactly how good or situational it is.
• Breath of Life doesn't seem too great to me in a meta with so many very strong offensive threats, but I some others disagree, so I'm mentioning it here anyway. In combination with the BoL sacred seal and possibly Aura the healing is more notable, but you're also sacrificing more skill slots for it.

3-4 star heroes who have these skills

Reposition is available from 3* Barst and Selena, Draw Back from 3* Nino and Sully and 4* Fae, Swap from Arthur, Stahl, and Subaki.
Rehabilitate is on 4* Lissa, Wrys and Priscilla.

You can get Moonbow from Palla, Odin, and Athena, Luna from Catria and Frederick, Draconic Aura from Camilla and F!Corrin, Bonfire from M!Robin and A!Tiki, Iceberg from Niles, Shanna and Lloyd.

You can get some of the good full rank passives from these 4* heroes:
Fury from Hinata, Death Blow from Klein, Triangle Adept from Roy, Desperation from Shanna, Vantage from Lon'qu and Reinhardt, Wings of Mercy from Cain, Swordbreaker from Abel, Lancebreaker from Narcian, Axebreaker from Laslow.

Resources to use

• As already mentioned, the wiki is great for looking up basically every piece of information about this game.
• There are a duel/damage and a mass duel (puts a unit with skills you choose up against every unit in the game and shows the results) calculator that are great for checking individual matchups and overall performance of heroes with skill combinations you are considering using.
• This chart lists units' stats with various convenient ways to filter/sort them.