Painting an elder dresser and headboard? My mother-in-law gave us a bedroom set from the 80's to...

My mother-in-law gave us a bedroom set from the 80's to use within our spare bedroom. I want to try and paint the set and get current hardware for it. I was wondering what type of paint to use. I devise there is a wood veneer over the top. There are also what look close to water spots and mischief on the surfaces of the pieces. This will be my first attempt at something like this, so any counsel will be appreciated. Thanks!
Answers:    If it is a veneer over MDF, you cannot sand the veneer as it will scratch. Been in attendance, done that. You can degloss it either by using steel wool or a product resembling, Paseo, a deglosser. Then use a stain blocking primer like, Zinsser. I prefer to any use a sponge roller or if possible, spray it on for the smoothest finish and to not hold to sand away brush strokes. Then you can either spray it or paint it near a regular latex paint. I usually seal the paint near a non-yellowing, UVB/UVA protecting polyurethane to make it easier to verbs and to protect the finish.