The paint in our bathroom is starting to peal and there are strange "brown spots" guessing from a shampoo or something (definitely not mold). What can I use to clean that up before repainting everything?
Use TSP. You can get it at HD/Lowes. Follow the precautions!
I'm assuming that you're painting over drywall, and not tile or plastic or some other substrate? Use a good stain hiding primer such as KILZ, and then paint over it with a premium paint such as Benjamin Moore Aquavelvet.
KILZ will cover the stain and prevent it from showing through again.
Using a premium paint will leave a thicker, more water- and chemical- resistant finish that won't wipe off the first time you clean the bathroom.