Clean the entire piece with a lint-free cloth, inside and out, to remove any stuck-on residue. Use a household cleaner that is appropriate for wood furniture.
Remove the hardware and drawers if your piece has them.
Sand all the surfaces of the piece with fine-grade sandpaper. It is not necessary to sand down to bare wood. Sand just enough to roughen up the finish so it will accept the primer.
Wipe down all surfaces with a damp cloth to remove saw dust. Allow the surface to dry completely.
Spray on a coat of primer to all surfaces that will be painted.
Apply two to three thin coats of interior paint using long, smooth brushstrokes. Allow each coat of paint to dry completely before applying the next coat.
Apply one to two coats of wipe-on polyurethane with a dry lint-free cloth. After the first coat is dry, sand it very lightly with steel wool and wipe away dust before applying a second coat.
Allow your piece of furniture to dry for at least 36 hours before replacing the hardware and drawers.