I've stripped and sanded the entire thing (except the legs, those are last), patched the missing chunks on the sides, and now I've primed the main part. I'm gonna paint it white with an oil-based paint (probably spraypaint) and leave the drawers the wood tone. They have some chips, but they were overall in much better condition than the rest of the dresser. We'll consider the wear and tear as "character". I was going to paint the legs white also, but I'm kinda digging the wood tone. The top is a plastic laminate or something, I decided it would be easiest to just leave it as is.
It looked like this before: