Sounds like you've got a good handle on this job already. If the product has a grain it's most likely wood, can you check a concealed area by trying to lift off a small piece of the veneer? Either way doesn't matter, the finish will be comparable. Use steel wool to sand any mouldings or detail (where sandpaper won't work). Apply 1 or two coats of undercoat, depending on coverage/ finish you require. Rub down LIGHTLY between coats of paint with steel wool; this is called denibbing- it removes the fine particles of dirt/ dust, loose bristles, runs etc that mar the paintwork. Finish in your selected topcoat. Eggshell is an old type of finish, similar to satinwood but (I think) more appealing. It's a little bit smellier than satinwood (higher VOCs) but is pleasant to use and goes on well. It's a little flatter than satinwood (does actually resemble an eggshell) and is tough. For painting i recommend painting all mouldings and detail trim with a good quality bristle brush; finish large flat panels with a small 4" roller- unless your brush technique is good. With this type of workn I would probably give two coats of the finish paint, for a real quality finish. It's very difficult to get a good, consistent finish in one coat only. Good luck with it. When dry give the units a rub with some spray furniture polish to even the sheen.