Disassemble
It will be easier to work with if you can take the furniture apart, or at least break it down in to smaller pieces. If you can't take it apart, you'll still want to remove any doors or drawers.
Sand
You'll want to sand the surfaces that will be painted, with 220 grit sand paper. Be careful not to sand too much, you don't want to sand completely through the veneer. You're just looking to remove the protective coating, and scuff up the surface a bit.
Wipe away the dust
Wipe the surfaces down with a Tack cloth to remove all the dust.
Prime
Prime the surfaces with a primer designed for laminate, or "glossy surfaces". Zinsser Bulls Eye 1-2-3 Primer/Sealer is one such primer, though somebody in the paint department at your local hardware store may be able recommend a better/different one.
Wait
You'll want to wait the full cure time specified in the instructions on the primer, before you start painting.
Paint
Apply 2 coats of paint (waiting the specified time between coats), with a foam brush and/or roller. You can use any type of brush you want, but a regular brush might leave streaks while a regular roller may leave an undesired texture.
Seal
Once the paint has fully dried, you can apply a polyurethane to help protect the finish. Apply 2-3 coats with a foam or soft bristle brush, sanding between coats with 220 or 320 grit sand paper (See What grit sandpaper should I use between coats of poly?).
Put it back together
Reassemble the furniture, and touch up any spots damaged during assembly.
No two whites are the same. In fact, sometimes two tins of the same color from the same company can be different! This is why if you are painting large areas, it is best to blend the cans together.
Best Answer
Point of order, the term enamel is a tricky one. For the purposes of this conversation, lets just say that enamel is the generic name for any paint that has had an additional hardener added to it. Finishes, outside the automotive world, don't really have enamel per say. Not important just clarifying.
You can go through the process of finding out what your finish is if you want to be thorough (there's any number of resorces on the internet on how to do this) but really when you're refinishing there's only two things to worry about-
A: will the new finish react with the old finish, and
B: will the new finish stick to the old one.
On the first point, its unlikely that you as DIYer (no offense) will have access to anything that will reactivate a descent wood finish, especially a factory finish. That means adhesion is your primary concern. Start with a thorough sanding with 280 to give the existing finish some tooth. If you find any areas where the finish is failing, sand or scrape off the loose bits. Blow off the dust and wipe with a tack cloth or rag dampened with turpentine. Now give it a good coat of a shellac based primer or sanding sealer. Do NOT use paints that have a primer included. They don't cover as well as regular paint, and they definitely don't prime and seal as well as dedicated primers. Plus they aren't shellac based which is what you want because it will stick to almost anything you put it on and will not react with anything you put on top of it. Best of luck!