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.
Alkyd paints are worse than a waste of effort- the water vapor permeabilty is much lower than that of acrylic, making it more prone to trapping moisture leading to blistering/cracking/flaking. It's relatively waterproof, but in places with numerous & rapid shifts in humidity it is much more likely to have problems than something highly vapor-permeable. Acrylic paints are about 5x more vapor-permeable than oil paints, so it dries quickly & completely. PVA & standard latex paints have comparable permeabilty (~5.5-6 perms), but will vary in other characteristics such as capillary draw, etc.
Gloss/semigloss acrylic is "naturally" mildew resistant, since it's waterproof to liquid water (won't wick water into the wall from a splashes or condensation), but fairly vapor-permeable- it dries both into the gypsum/wall-cavity, and the room as humidity conditions in each vary, and allows the wall cavity to dry toward the room (when the room humidity returns to normal) without causing seperation at the paint/wallboard boundary.
But nothing will stay mildew-free if the room relative humidity is allowed to linger at 70% or higher. Even the active ventilation the room RH runs near 100% whenever there's a shower in progress, but with a decent bathroom fan or heat recovery ventilator (under dehumidistat control, if you like) it's manageable. If continuous flow or dehumidtstat control the flow rate of the ventilation doesn't matter a huge amount- a bigger fan will lower the humidity quicker, but none will keep up with a full-on shower, yet even 7-10cfm continuous flow would be able to sufficiently purge humidity in most residential bathrooms, (even though the IRC calls for 20cfm, if continuous flow.) With human-operated ventilation performance is less guaranteed, since the fan is flipped on/off at semi-random intervals whenever the user
thinks it's done it's job. Going with a 150cfm fan might be an improvement over a 50cfm fan if manually operated, but dehumidistat control or continuous low-flow is more reliable, and you can do with the smallest, quietest of fans.