Remove the drawers from the dresser and lay them out on the ground. Unscrew the dresser knobs with your screwdriver, and keep the knobs in a safe place where they won't roll away.
Fill any holes in the drawers with wood filler. Dip a paintbrush into the wood filler and stick it into the holes. Use a putty knife to smooth out the surface once it dries. Filling in the holes prevents air bubbles from forming beneath the coat of new paint.
Sand down the drawers with your sandpaper strips. A rough sandpaper grit of 50 to 80 is best for removing old paint from wooden furniture, according to Repair Home. Sweep away dust or leftover sand or grit from the sandpaper with your dust rag.
Coat the drawers with primer. Once dry, paint your color coat on top. Apply two coats of paint and a finishing coat, such as a high-gloss or a water-resistant coating.
Allow the drawers to dry completely. Replace the knobs by screwing them back in. Put the drawers back into the dresser structure.