Whether you're painting over stained and varnished cabinets, trim, doors or furniture, the process is essentially the same. The smooth surface must be sanded and the wood primed or the paint won't stick. Done properly, paint over stain and varnish can make a dramatic difference and will be durable, washable and resistant to chipping. The same process works for both interior and exterior surfaces as long as you use the right primer and paint.
Clean the surface thoroughly with a de-greasing detergent. Use dishwashing detergent or trisodium phosphate (TSP) cleaner. Scrub it thoroughly using a scrub or sanding sponge to remove wax, oil and dirt.
Sand the entire varnished surface until it is dull. Use either a medium-fine sanding sponge or 120-grit sandpaper at first, then a final sanding with 150- or 180-grit sandpaper for a very smooth finish. Wipe or vacuum it to remove all the sanding dust. Dampen a clean rag with mineral spirits to pick up dust from grooves and corners.
Prime the varnished wood with tinted shellac or oil-based primer. Use only exterior oil-based primer for exterior surfaces: don't use tinted shellac primer outside. Apply the primer with a lambswool or mohair roller for larger surfaces, and either a China bristle or good quality synthetic brush for brushed-on primer. Allow it to dry according to label directions.
Paint the wood with at least two coats of either water-based (latex or acrylic) or oil-based paint. For a very smooth surface, allow at least eight hours for the first coat to dry, then sand it lightly with 150- or 180-grit sandpaper before applying the second coat.
Finish the project with two coats of varnish for extra durability. Use acrylic varnish over water-based paint, and oil-based varnish over oil-based paint. This step should not be necessary if you used high-quality paint but may make some surfaces, such as tabletops or shelving more washable. Note that oil-based varnish may tend to yellow the underlying colour.
One coat of primer should be sufficient for most surfaces and an additional coat will not increase paint adhesion if you are using a good-quality primer. For intricate surfaces and furniture, some spray-can paints can be used without a primer. You will still need to clean and sand the surface carefully, then follow the spray-can label directions for application.
If the surface is very smooth already, you may be tempted to use a wipe-on "liquid sander" product instead of sanding. These are very convenient but may not etch some surfaces enough to the primer to adhere well. To cover all bases, sand the varnished wood lightly and follow with liquid sander.