Clean the desk with warm, soapy water and a sponge. Scrub away any dirt. Allow to dry. Evaluate the condition of the desk. If it is unpainted, you'll only need to sand off any remnants of varnish using 80 grit sandpaper and proceed to Step 3. If the desk is painted, but the paint shows no signs of peeling, and the desk is in overall good condition, you can choose to paint over the existing paint. More typical is the old school desk with multiple layers of peeling and flaking paint. If this is the case, you'll need to take further action.
Strip the old paint from the school desk using paint remover. If there are many layers of old paint, use a brushable paint remover and let it sit the amount of time indicated on the can. Remove the paint remover by scraping it off with a metal scraper, unless you have purchased the type of paint remover that can be removed with water. After you have removed the majority of the paint with the scraper, use medium-grade steel wool to remove the bits around the joints. Make sure that you are in a well-ventilated area when performing this step, as paint removers give off toxic fumes. Sand the desk by hand with 80 grit sandpaper to remove any remaining bits of paint and to provide a good surface to the paint to cling to.
Fill in any holes or deep scratches with wood putty. Use a razor blade (Reference 1) or the scraper to smooth the surface, and allow to dry. After the wood putty is dry, sand in the direction of the grain of the wood with a 120 grit sandpaper to make it smooth and even with the wood.
Spread a tarp or old newspapers underneath the desk. Open the can of paint and stir it vigorously. If the paint looks as though it will coat the desk too thickly, add a bit of turpentine to the paint to thin it out a bit. Using a soft, flat bristle brush, apply the first coat of paint to your desk. Allow the paint to dry for one day. Go over the desk with a very fine 00 grit piece of sandpaper, and apply a second coat of paint. Allow the final coat of paint to dry for at least 36 hours before using the desk.