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.

Now you know why varnish is rarely used anymore. If it is pure varnish/shellac, then damage to the wood is likely. Before sanding and refinishing, the nuclear option is to try to rub it out with denatured alcohol. The alcohol will dissolve the varnish and redistribute or remove it. You can then touch up the varnish carefully.

If it is lacquer, then use lacquer thinner. Remember, varnish and lacquer are not the same.

This is somewhat successful in some stains.