Use some nail polish remover to see if the paint on your shower walls and ceiling is a latex or an oil.
You generally don't need to prime over paint. If you have a latex paint you can paint over it with another latex paint after giving the wall a good cleaning and allowing time to dry. Same thing for oil based paint. Provided that you don't have a semi-gloss or gloss paint now, you can just paint over what you have with the same kind of paint without having to sand or roughen the surface.
If you have an oil based paint now, and you want to paint with a latex, then you normally need to prime the oil based paint with an oil based primer, and then paint over the oil based primer with a latex.