How to paint over varnished wood cabinets:
(1) Remove cabinet doors and all hardware (door/drawer handles, pulls, hinges, etc.). Consider whether you want to replace hardware with new; if not, clean hardware well and put aside.
(2) Empty cabinets completely. Clean thoroughly inside, and remove any old contact paper.
(3) Using #50 or coarser sandpaper, sand off as much of the varnish as possible. It’s a very good idea to have a dust mask, as the sanded-off varnish dust will coat the inside of your nostrils otherwise, which is nasty and probably not good for you. You will find that the varnish gums up your sandpaper often, rendering it ineffective. Have a generous supply of sandpaper at hand. If you value the skin on your hands at all, wear gloves while sanding.
(4) After you’ve finished with the #50 sandpaper, repeat the process with #100 sandpaper. If the wood has scratches, gouges, etc., that need repair, fill them (in between sandings) with wood putty and allow the putty to dry - about 15-20 minutes for shallow fills, longer for deeper repairs. When you’ve finished sanding with the #100 sandpaper, all repairs should be smooth to the touch, so that they will not be noticeable when painted over. Also, your wood surface should be nice & smooth as well. It isn’t necessary to sand all the way down to bare wood, but at least remove enough varnish that the surface is matte and smooth. Wipe surfaces clean after sanding.
(5) Paint the surfaces with an oil-based primer. Since this stuff is really difficult to remove from brushes, I recommend using cheapfoam brushes, and throwing them away after each use. Allow the primer to dry, at least an hour or more, depending on manufacturer’s recommendation.
(6) Using #150 sandpaper, sand the primered surfaces lightly until smooth. Wipe surfaces clean after sanding. Next, apply a thin coat of acrylic-enamel paint (you may get smoother results if you add paint conditioner to the paint). Use the best brushes you can for this, and try to minimize the appearance of brush strokes. Allow to dry at least 6 hours - overnight, if possible.
(7) Repeat step 6 as desired - I recommend 4 coats of paint total, especially if you have thinned the paint with conditioner and/or for lighter colors.
(8) After final coat of paint has dried, reinstall hardware and rehang doors. You may wish to apply foam or felt spacers between door/drawer surfaces and cabinet surfaces, as the painted surfaces may stick together until the paint is fully cured.
(9) Enjoy the new look of your cabinets!