Remove the hardware from the cabinet using a screwdriver. Set the hardware and screws aside for safe keeping.
Clean the cabinets with soapy water and a rag, and clean again with a degreaser such as TSP. Wait for the cabinets to dry.
Sand all surfaces to be painted with an 80-grit sandpaper to rough up the finish.
Prime the cabinets with a melamine primer. Use whatever painting utensil the manufacturer recommends, whether it's a natural bristle brush or a foam brush. Wait for the primer to dry.
Sand the primer very lightly with 240-grit sandpaper.
Paint the cabinets with melamine paint. Apply the paint in slow, even strokes. Melamine paint should even itself out as it dries, but you can ensure this by painting with brushstrokes going all in the same direction.