Place the dresser over plastic sheeting. Remove all hardware and drawers. Cover any part you do not want to paint on the dresser with painter's tape and plastic.
Sand the dresser and drawer fronts with an orbital sander or a sanding block. If the previous finish is peeling or bubbling, you need to remove it. Otherwise, sand enough to smooth out imperfections and create a rough surface that the paint will adhere to better. Wear a mask as you sand to prevent yourself from inhaling any chemicals sanding may release.
Use a paintbrush to paint white primer onto the dresser. Move the paintbrush in the same direction as the wood. Paint the face of the drawers and, if desired, the inside. Let primer dry.
Use a paintbrush to paint white paint onto the dresser, moving in the same direction as the wood grain. Paint the drawers. Add paint layers as needed to get the desired color, letting paint dry in between coats.
Spray a coat of polyurethane on dresser to protect the paint from scratching.