Remove the drawers and all hardware, and set it aside.
Strip the table top of the vanity if your vanity is painted, varnished or if it is stained a color you don't like. Use the appropriate finish remover and apply it according to the manufacturer's directions. If the wood is unfinished, stained and you like the color, leave it. You don't need to strip the legs, drawers or other surfaces.
Sand every surface of the vanity with 150 to 180 grit sandpaper, including any drawers and the drawer edges. Wipe the entire vanity with a tack cloth to remove dust. Cover the bottom parts of the vanity with painter's tape and newspaper to protect it while you work on the top.
Put on protective gloves. Apply a wood stain to the table top of the vanity using a rag. Rub the stain on the oak in the direction of the grain to work it into the wood until the entire top is coated. Give it a second coat after the first coat dries, which usually takes between 4 to 24 hours depending on the stain and brand. Let the second coat dry.
Rub some dark shoe polish on a rag and rub it onto some areas of the wood-stained top of the vanity. Imagine where it would have been worn from age and time, such as the corners or the surface where it would be used the most. Rub the shoe polish on it to discolor it and wipe up most of the excess.
Sand the top, if desired, to further distress it.
Paint the entire rest of the desk, including drawers. White is ideal for shabby chic, but a pale blue, pink or mint color will suffice. If your drawer pulls are wood, paint those as well. Allow the paint to dry. Replace the handles.
Sand all the painted surfaces with fine grit sandpaper. Switch to an 80 to 120 medium grit paper and sand down the corners, edges, handles and around the edges of the drawers. Sand it until you reveal a little of the old surface below.