Refinishing your under-the-sink cabinet is not much different than refinishing the cabinets that hang above your counters. Watch out for the plumbing. It is a bit more difficult to maneuver yourself around the plumbing inside your cabinet, but it can be done.
Take It Out
The first thing to do is to remove everything from the cabinet, including any shelving. That should leave your cabinet empty, except for the plumbing.
Cover the Plumbing
Wrap a plastic bag around the plumbing. Use duct tape to seal the bag. Doing so will ensure that you don't get any finish on the plumbing itself.
Cover the Floor
Using either a large piece of plastic or a drop cloth, cover the floor where you are working. Doing so will ensure that most of the mess stays in a small area, saving clean-up later.
Take Doors and Handles Off
Remove the doors from their hinges using your screwdriver. Lay them down onto the plastic, and remove the handles from the doors.
If you plan on using those same handles when your cabinet is done, lay them aside. If you have bought new handles, you can throw away the old ones.
Wash out your cabinets inside and out with warm water and cleaner. Rinse with cool water.
Fill The Holes
There are bound to be holes in your cabinet from screws and nails. Use wood putty or wood dowels to fill the holes.
Sand your cabinet to an even surface. It doesn't have to be completely smooth, but it should be even and nice.
Use either a handheld vacuum or your regular vacuum attachments to clean dust from sanding.
Now you must choose between varnish and paint. Whichever you choose, apply it in even strokes, catching any drips so that you end up with a clean-finished product.
Now replace the doors, knobs, and handles and allow the finish to dry before using the sink.