Sand everything smooth paying special attention to the peeling areas. Clean everything to remove the dust and now use a good stain blocking oil base primer. A good acrylic semi-gloss paint can be used for the finish but an oil base is better. Go to your local paint store, not a hardware store, and ask for their best cabinet paint.
It will take one maybe two coats of both primer and two or more coats of paint, depends on application method.