The first step is to use the right kind of wood. As we discussed before, birch dowels are far preferable to oak ones. Oak has an extremely coarse grain that, on model-sized parts, is extremely difficult to fill completely. Balsa is worse. The most commonly-available modeling wood (other than balsa, which I don't recommend for serious scale ship modeling) is basswood. It's a pretty good wood for painting; it's easy to carve, and the grain, though a little fuzzy, is fine enough to respond to a good filler. The Lowe's store a few blocks from my house carries a supply of nice poplar, which isn't a particularly attractive wood for clear finishes but is great as a basis for paint.
If you're willing to look a little harder, very hard, close-grained woods like maple need virtually no filler before painting. And the prize woods among scratchbuilders are boxwood, pearwood, and holly, which have scarcely any visible grain at all. But they're harder to find, and expensive.
When furniture and cabinet makers confont this problem they generally use "paste filler," which is a thick, rather muddy-looking substance that's rubbed into the grain before the finish coats are applied. That's pretty brutal stuff for model building, though. The "sanding sealer" sold in the flying model airplane section of the hobby shop seems to be good stuff - though I haven't tried it myself. On poplar, which I use for "sea" bases for waterline models, I've gotten good results with artist's gesso (from the art supply department). Two coats of that stuff, which is a thick, creamy white, water-soluble liquid applied with a brush, do the trick. (I learned the hard way, though, to put it on both sides of the board. If it's applied to only one side, the board will warp.) Paint stores and home centers also sell primers that work nicely. Be warned, though: be sure the primer is compatible with the paint you're going to use for the finish coat.
Any of these tricks, in conjunction with some super-fine sandpaper and elbow grease, will work fine. Good luck.
Youth, talent, hard work, and enthusiasm are no match for old age and treachery.