Remove the doors, hinges, handles and any other hardware from the vanity. If there's a drawer, pull it out. There's no need to unscrew the drawer fronts from the drawer box.
Sand the entire vanity, always sanding in strokes with the grain instead of across it. Sanding across the grain causes scratches that will be visible after you apply the stain. Remove the sanding dust with a soft, lint-free rag such as cheesecloth.
Apply the stain with a soft brush, allow it to penetrate for 10 to 15 minutes, then remove the excess with cheesecloth or clean rags. Work in sections, such as one side of a door at a time.
Allow the stain to dry for at least 16 hours in a warm, dry, well-ventilated area.
Apply at least three coats of varnish to the vanity with a 2- or 3-inch brush, sanding lightly between coats with 180-grit sandpaper or a fine sanding sponge.
Wait for the final coat of varnish to dry for at least eight hours before reinstalling the cabinet doors and drawers.