Best Mop - What do you use?

We have 3 dogs and a lot of hard floors and live in the PNW. I know I worry too much about the floors and I try to let a lot go. It is what it is with pets.

The other day I swept, then vacuumed the same floor, used a steam mop, and then used a swiffer and I know the floors are still dirty, even though they "look" clean. The pads on the mops were dirty but not THAT dirty and I changed them several times. I know they are still dirty because I spilled something on the floor and wiped it up with sanitizer and a paper towel just in one little space and the paper towel was FILTHY. This was so defeating because I worked so hard to get clean floors and it's a WORK OUT. I have to split the task over two days.

So long story short, whats the best mop to use? I want to work smarter, not harder.

Thanks!