Remove the dresser drawers. Unscrew the handles with a screwdriver and remove them.
Sand the surface of the drawers with 220 grit sandpaper, following the grain of the wood. Wipe the drawers with tack cloth to remove the dust.
Apply one coat of high gloss interior paint or one coat of wood stain in your preferred color. Use a foam paintbrush, and follow the wood grain. Apply the colorant in even, overlapping strokes. If you are staining, let it sit for two to three minutes, then wipe the excess stain off with a rag.
Let the first coat dry for two to three hours. Apply a second coat, if you want a darker color. Let that coat dry.
Apply a coat of spray-on polyurethane varnish to the drawer surface. Let the drawers dry according to the directions on the varnish.
Install new drawer pulls in the holes where the old pulls were. Screw them into place with the screwdriver.