Couple more factors to consider before you paint. Whenever repainting old work, preparation is the most important step in the process. Old finishes need to be very clean, all soaps, residues removed and surfaces sanded or chemically deglossed. Be aware, any paints before 1978 may contain lead and should be tested and handled properly to avoid any health risks.
Surfaces should be primed, if there is any bare wood, I like pigmented shellac, like Bin's Bullseye primer. Over old paints, a good high hiding primer like Kilz if fine. Always do a quick buff sanding and damp wipe down of the dried primer before applying top coats. In bathrooms, I usually prefer a latex enamel semi-gloss paint. The choice of full gloss or semi-gloss is simply a personal preference. An enamel give you a very hard finish that cleans well and will hold up for years. Always use a good quality brush. Please, no bargain basement $0.99 specials. A quality job comes from proper prep, quality products and the right tools. Good luck.