Sand the dark wood furniture with fine-grit sandpaper. You are just roughing up the finish of the furniture, not sanding it down to the bare wood.
Vacuum the furniture to remove the sanding dust. Wipe the furniture down with a damp cloth to remove any leftover dust.
Place the piece of furniture on a drop cloth in a well ventilated area. Apply paint primer to the piece with a paintbrush and a foam paint roller. Use the paintbrush to reach into the corners. The foam roller, if used properly, will spread the primer without leaving brush lines. Let the primer dry per manufacturer directions.
Paint the furniture with white latex paint in an eggshell or semi-gloss finish. Flat finishes pick up dirt quickly, and high-gloss finishes show painting and surface mistakes easily. Coat the furniture with several light layers, and give each sufficient time to dry before applying the next layer. After the final coat, wait for the paint to cure for several days before using the furniture.