Scrub the paneling clean with a sponge and grease-cutting detergent, such as dishwashing liquid. Rinse it and allow it to dry.
Fill holes in the paneling with spackling compound. If the paneling has a pronounced wood-grain, use your finger to apply the compound to the hole directly instead of using a putty knife, which will create visible flat spots.
Sand the paneling to dull the surface, using 120-grit sandpaper or a sanding sponge. Wipe away the dust with a damp, lint-free rag.
Apply 2-inch painter's tape to any adjacent surfaces you're not going to paint, such as the ceiling and trim.
Prime the paneling with solvent- or water-based stain-blocking primer. Use solvent-based primer, such as shellac- or oil-based, for the best results. Cut in around the edges of the paneling with a 2- or 3-inch brush, then roll the rest of it. A 1/2-inch lambswool roller cover holds more paint and will get paint into grooves in the paneling more efficiently than an inexpensive synthetic roller cover. Allow the primer to dry.
Caulk visible gaps and joints in the paneling with white caulking compound. Run a bead of caulk down each gap. Immediately smooth it down with your finger or a damp cloth.
Paint the paneling with good-quality interior latex paint.