The procedure for fixing your wall is pretty straight forward.
Step 1) Remove the loose paint and prepare the area for patching. This involves scraping, sanding and cleaning.
First remove all loose paint and any other loose material. A stiff broad putty knife works on smooth walls or surfaces with little texture. Ruff textured surfaces might require the use of a wire brush after scraping. All loose paint and any loose wall material needs to be removed. Start with a careful scraping to expose the underlying surface then progress from there if the peeling area needs it.
Sanding might be needed depending on the amount of layers of paint that are peeling, the overall condition of the area and if the wall has texture or is smooth. Generally sanding is needed on smooth walls in order to blend the edges of the paint after scraping or if extensive patching is needed on textured walls. Use 120 grit sand paper to smooth these areas. Sand a little beyond the affected area, this will help the patching compound stick better.
After sanding it is a good idea to remove the dust with a vacuum and wash the area. Nothing special, a soft cloth or sponge and warm water will work. If the walls have any mold use bleach and water, 50-50 mix. An excessively dirty wall will need to be thoroughly washed before priming or painting. In this situation use Trisodium Phosphate or an equally strong cleaner and rinse well.
Step 2) Priming and patching is the next step. If the area doesn't require extensive repair just prime it with a quick drying oil base primer. Kilz Original oil base or Zinsser Cover Stain works the best. If your needs are modest both products are available in an aerosol can. The amount of patching depends on the damage.
I like to use a setting type drywall mud for most repairs. This is a powder with variable setting times. Unfortunately it is available in 20 pound bags. One bag will patch a lot of areas. I don't know if it available in smaller quantities, I only purchase the large bags. If you decide to use this type of patching compound, purchase the 45 minute setting time. It dries quickly but not too quickly.
All patched areas need to be primed before painting. This will prevent flashing of the patches plus help the paint stick. Priming the entire wall is necessary if the existing finish is an oil base and you want to use an acrylic or latex over it.
Step 3) Now it's time to choose your paint. Your choice is oil base or water base. Both have their positives and negatives. Most of the time, I prefer to use 100% Acrylic paint.
Use an oil base paint if the ceiling and walls have several coats of oil base already, if not use the acrylic. Whatever type you choose purchase a name brand, such as Benjamin Moore or Kelly Moore, and purchase their best. Cheap paint will just peel again. Also use either a satin or semi-gloss sheen, semi-gloss will deal with high humidity better.
To help prevent any peeling in the future use an exhaust fan or open a window during showering.