Songbird:
You've gotten a lot of advice, and some of it is more or less right.
What you should do is this:
1. Kill the mildew that is on your paint now. You CAN use bleach to do that, but mildew is not at all hard to kill, and even washing the walls and bathroom ceiling with a mild solution of TSP will do the job.
PS: There is a common misconception that it's recommended to clean walls with TSP prior to repainting because TSP is such a good cleaner, or cuz it's such a good degreaser. In fact, TSP is a lousy cleaner and a lousy degreaser. TSP is sold in the painting aisle of the hardware store rather than the cleaning aisle because it will etch the gloss of drying OIL based paints like true linseed oil based paints, thereby roughening the paint surface so that a subsequent coat of paint will stick better. However, so far as I know, it only etches the true drying oil based films (like linseed oil based paints or Tung oil varnishes) that have since been replaced alkyd paints and polyurethane "varnishes". So far as I know, TSP does not etch alkyd or alkyd based polyurethanes; it only etches true drying oils like linseed oil based paints, Tung oil varnishes and artist's oil based paints made with walnut or poppyseed oil. Also, cleaning a latex paint with TSP prior to repainting won't etch the gloss. It'll only kill the mildew on the old paint.
2. Rinse any cleaner you used off the walls and ceiling to prevent it from interfering with the adhesion of the paint.
3. Paint with a paint specifically meant for bathrooms like Zinsser's PermaWhite Bathroom Paint, which you can buy at Home Depot. This paint is a tintable latex paint that has a mildewcide in it that is drawn to the surface of the dry paint film because of it's high solubility in water. At the surface, the mildewcide kills any mildew spores that land on the paint before they have a chance to grow, thereby keeping the bathroom mildew free.
http://www.zinsser.com/lowes/pwint.html
PS: I know what I'm talking about. I own a small apartment block in Winnipeg, Manitoba, and none of my 21 bathrooms have any mildew on them. I assure you that my tenants don't bother using the bathroom fans more than they can help it, and there's nothing I can do to make them. You can see my building at http://www.ilos.net/~nkelebay
I seal my bathroom grout to prevent mildew growth there, too.
PS#1: The worst paint you can use to prevent mildew in a bathroom is a true drying oil based paint (like the linseed oil based paints that were last available in the 1980's). The reason is that the linseed oil itself forms a food source for mildew. Alkyd paints (which are basically "clumps" of modified oil fatty acids) are less susceptible to attack by mildew, and alkyd based polyurethanes are less susceptible to mildew than alkyd paints.
PS#2:
Another common problem is latex paints PEELING in the wettest areas of bathrooms, like right above your head on the shower ceiling, or high up in the corners of the room where it's coolest (due to heat loss) and condensation occurs. This is because latex paints are generally made with one of two different kinds of plastics; plexiglass or a polyvinyl acetate (which you know better as "white wood glue"). The latter PVA plastic is used in general purpose primers and budget priced interior and exterior paints. PMMA (or polymethyl methacrylate or plexiglass) is used in concrete primers and kitchen & bath paints and better quality interior and exterior paints. The problem is that inexpensive PVA paints will loose both their strength and adhesion if they get wet (even after they've been dry for years), and it's this loss of adhesion that causes the peeling on bathroom ceilings. Painting over the PVA paint with a better quality acrylic paint will keep the PVA dry and solve the problem.
Also, someone said that oil based paints "peel". This is only true of oil based paints that are applied over wood outdoors. The reason why is that oil based paints get progressively harder and less elastic the longer they're exposed to oxygen in the air, whereas wood is a natural material that swells and shrinks with changes in it's moisture content most usually due to changes in the seasonal humidity. Consequently, eventually the oil based paint is no longer able to stretch and shrink with the wood, and the result is that it begins to crack and peel. THIS is the single biggest reason why exterior latex paints offer the longest lasting protection over wood outdoors. Latex paints don't loose their elasticity with age.
PS#3:
Because oil based paints form harder films than latex paints, they provide better service on WORKING surfaces, such as floors, shelves and furniture. If you paint a floor or cupboard shelf or window sill with a latex paint, dirt will become embedded in it's softer surface, and it will soon start to look very dirty. By using a paint that's harder than the dirt, you prevent that from happening and the result is that oil based paints will stay looking good on a floor for a lot longer than a latex paint will, even a latex paint intended for use on floors.
PS#4:
The VAST majority of people know very little about paint, even those that working in paint stores. The only reason why they get away with it is that the rest of us know even less. Educate yourself, and one of the best places to do it is the Paint Quality Institute's web site at:
www.paintquality.com
The Paint Quality Institute was established by the Rohm & Haas Company who are the largest producer of the acrylic resins used to make latex paints, acrylic floor finishes, acrylic grout sealers and acrylic plastics (like Plexiglass) in North America. The purpose of the Paint Quality Institute is to educate painters, architects and home owners of the benefits of choosing higher quality paints. That is, they are the people that make the acrylic resins which companies like Pratt & Lambert or Benjamin Moore buy to make latex paint from. Consequently, the chemists at Rohm & Haas actually make acrylic resins. The chemists at Pratt & Lambert or Benjamin Moore are mostly just employed to do quality control work on the raw materials they buy and the paint they produce from them.
PS#5:
Someone said oil based paints take a long time to dry. A paint made using boiled linseed oil as the binder will take a day or three to dry. However, those oil based paints are no longer commonly available. Nowadays ALKYD paints are made from MODIFYING the fatty acids of vegetable oils so that their degree of unsaturation is increased tremendously. An alkyd "resin" is nothing more than a clump of "fatty acids on steroids". As a result, modern interior alkyd paints dry to a much harder film much faster than the old boiled linseed oil paints ever did. Nowadays, an interior alkyd paint is dry to the touch in several hours as compared to several days like a linseed oil based paint would take to "dry".
PS#6: Someone mentioned "waterborne" paints.
These are not latex paints, but they wash up with water just like latex paints though. I've used them and I've very impressed with their hardness and durability. Also, I don't know anything about the chemistry by which they form films, and can only offer an educated guess in that regard. Still, you don't need a hard film on a wall or ceiling like you do on a working surface like a floor or counter top, so unless the waterborne paint is intended for use in a bathroom and has mildewcides in it, you'd be best off using a paint intended for use in bathrooms like the Zinsser product.