So, I have decided that I should probably repaint my bathroom. The paint over the bath tub is peeling off, and I’m pretty sure the moisture won’t be good for the wall behind it. What sort of paint should I use? How much? How do I remove the old coating of paint?
IANAP, but from what I’ve been told by painters and read in decoration mags, and from what I learned making paint for 5 years,
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you don’t need to get rid of old paint. You need to prepare the surface, the same way as you would do it for any other room. The specific preparation depends on the state of your wall: if the wall is noticeably bumpy you may want to even it out somewhat. You should clean out any mold but this is true for every room unless you live in the Sahara.
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many brands of paint include anti-mold. The company where I worked isn’t a “final brand”, our products were sold to paint brands who would pour the stuff into smaller containers and (optionally) add color, but we already put in biocides (anti mold and bacteria). Some of them asked us to run a biocide-behavior test which took several days and send them the results: if biocide activity was low, they added more. Kinds of paint that have been designed for bathrooms and kitchens are usually marked so: it’s not just a marketing gimmick, they have more antimold.
To remove the old paint - peel off what comes easily and scrape/sand the rest.
Prime the walls and finish with two coats of a “kitchens and bath” paint. This stuff’s formulated for damp environments and has some anti-mildew ingredients.
Don’t skimp on the scraping/sanding preparation. The finished job will only be as good as the prep work, and poor prep will only lead to heartbreak when the paint peels off two weeks later.
For a typical-size bathroom, you’re probably in that gray area of needing a smidge more than a quart, but less than half a gallon. Yeah, it’s a pain.
Go to a local paint store and chat them up - they’ll be able to set you up with the appropriate primer and such.
This bears repeating.
After you scrape/sand, you should clean the remaining paint very well. The standard product is called TSP, you mix the powder with water and scrub. It is a strong chemical, so have ventilation, gloves and goggles. The stronger and cleaner the surface, the better your primer and paint will stick.
If the current paint has a gloss finish, it will need to be ‘de-glossed’, either by sanding, as mentioned before, or by using a chemical de-glosser. I forget what it is called, but you can get it at any paint or hardware store. I usually go the sanding route. If you don’t do this, the new paint may not adhere well.
Antimold paint can be hazardous to breathe. Wear a chemical-certified mask, ventilate the area thoroughly, and of course remove pets from the home (especially birds) until the smell is gone.
Sailboat
If the paint’s peeling, then you need to strip it all off. As a general rule of thumb, if you can feel a blemish with your fingertips, it will still show once painted. Paint over peeled paintwork and the old paint/plaster boundaries will show. If you really don’t want to strip the whole bathroom, then just strip the peeling paint and feather in the edges with sandpaper.
I’ve had a great deal of success with specialist “kitchen and bathroom” paint, even in bathrooms that started off mouldy and peeling. Scrape off any peeling paint, fill any blemishes with plaster skim (much easier to sand flat than repair filler) and then paint. You can seal the plaster before painting if you like, but I didn’t bother with my current bathroom, and seem to have got away with it. Water from the shower head runs down it daily, and after a couple of years of this it shows no sign of mould or peeling. It’s easy to clean, and looks just like any other emulsion finish.
Make sure that when you’re scraping off the peeling paint to do a through job. Use a puttty knife to make sure that there isn’t a bubble that hasn’t peeled yet. Then scratch the rest of the glossy surface with a light grade sandpaper. Then clean with TSP. Then procede with with sealing and priming, patching cracks with Spackle, etc.
Your paint dealer will suggest a type of paint to your needs.
Have fun! 
Zinsser makes a product called Perma-White interior paint which is guaranteed mildew proof for 5 years. Personally, I love their stuff-top shelf.
If your bathroom was done with ordinary gypsum wallboard instead of moisture resistant wallboard, the problem may recur. One solution, aside from replacing all the wallboard, is vapor barrier paint. Many folks have used it in bathrooms, and it might be right for you. Consult your supplier.