What type of primer and paint are better used in a bathroom?
Ideally, if it's standard sheetrock, I'd suggest removing it. You normally want to use 'greenboard' (sheetrock with a waxed paper layer) or, even better, the paperless wallboard products like Densarmor (they use a fiberglass layer rather than paper). The paper is what the mildew will like to 'eat' so removing as much of that as you can, the better.

Barring that, they do make paints specifically for bathrooms. They usually have some sort of anti-mildew additive. 

Finally, be sure the bathroom is properly vented. That, alone, is the single best solution. If there is no ceiling fan, invest in that first. 

To answer the specifics:

Latex is water based, so that's the same thing. And is what you want to use--specifically ones labeled for bathrooms and high-moisture areas (ie, they have the mildew preventative additive). You COULD use Oil based paint, and that has historically been something to use in a bathroom, but latex paints have come a long way and Oil based interior paint is now a rarity and likely not what most DIYers would want to use anyways (it's a bit more finicky and smelly to apply and a lot more messy to clean up).