Visit to Bordeaux (3 Days)

Visiting Bordeaux for 3 days in late November - plan to spend a day in the city and the rest split between Left Bank (Médoc) and Right Bank (Saint-Émilion/Pomerol).

Looking for château recommendations across a range:

• A couple of "must visit" bucket-list stops, even if pricey
• Solid mid-tier estates that do a great tasting/tour experience
• Under-the-radar producers making genuinely good value Bordeaux - I'm not chasing labels

On budget: I typically buy in the $50–150/bottle range, with the occasional $200+ splurge. So recs across that spread would be great, not just the trophy names.

Also curious:

• How far ahead do I need to book appointments for the châteaux worth visiting?
• Is self-drive realistic for hitting both banks, or is a driver/tour worth it for 3 days?