Remove the veneer from the cabinet. Depending on the glue that was used when the veneer was applied, you may be able to run a heat gun over the surface and scrape the veneer away with a chisel. Don't dig into the wood itself. You can also sand the veneer surface off the cabinet. When doing this, keep in mind that the veneer is very thin. You don't want to damage the wood underneath the veneer.
Wash the surface of the cabinet with a cleanser that contains mineral spirits.
Fill any nicks in the wood with wood putty. Sand down any areas that you've patched, or that are rough.
Paint the bare wood with primer. Go with the grain of the wood when priming the cabinets, using long brushstrokes and covering 3 to 4 inches of the wood at a time. Apply the primer evenly, and wipe up any drips with the brush. If the cabinet has any intricate designs, use small brushes to apply the primer in these areas. Allow the primer to dry.
Paint the cabinets with the top coat of paint. Go with the grain of the wood when painting the cabinets, using long brushstrokes and covering 3 to 4 inches of the wood at a time. Apply the paint evenly, and wipe up any drips with the brush. Use a small brushes to apply paint in areas with detail work. Allow the paint to dry.
Touch up any spots where the paint is missing, or add a second coat of paint if desired, using the same method that you used to apply the first coat.
Reinstall the doors, handles and doorknobs on the cabinet.