Paint can stick to most surfaces, even varnished pine, as long as steps are taken to adequately prepare the area beforehand. Paint adheres best to rough materials because they offer better bonding, but varnished surfaces are smooth, glossy and slick, so sanding and elbow grease are necessary to prime these types of surfaces to receive a new coating. Take the time to prep your surface appropriately for the best, most professional results.
Sand the varnished pine with 180-grit sandpaper. Use enough pressure to scuff the surface lightly, but not so much that you remove the varnish entirely. Sand the entire piece evenly, until the original shine is dulled and a light, greyish dust coats the surface.
Wipe the surface with a tack cloth to remove the sanding dust. Turn the cloth frequently to keep a clean area against the surface of the item.
Coat the piece with oil-based, stain-blocking primer, following the manufacturer's instructions and applying an even coat with a paintbrush. Allow the primer to dry completely.
Sand the primed surface with 180-grit sandpaper very lightly, just enough to roughen it slightly. Use a tack cloth to wipe the dust created by the sanding from the piece.
Coat the primed surface with oil-based paint and the paintbrush using long, even strokes. Allow to dry and add a second coat if primer shows through the first coat or if the coat is uneven.
Oil-based primers work best on previously varnished surfaces. The primer will bond to the sanded varnish and provide an excellent base for the finish coat while holding back stains and blemishes that may try to bleed through.