Go to a good paint store, get their best PVA primer for the drywall. That is really your best first step, and assures future success.
For many years, the only thing I have painted bathroom walls with is top quality latex, semigloss for the walls, gloss on the ceilings and woodwork. I like Behr. Here in San Diego, I like Frazee. Just go to a good store, and use their best paint.
I think oil is a waste of effort. You can't really even buy a good quality oil base paint for residential use anymore, and any whitish oil paint will yellow.