There are all kinds of products available to fill nail and screw holes in wood. Cut wood plugs, wood fillers, ready patch... all good options. But my favorite method is one I can make myself, with two ingredients I always have. It's cheap, reliable, and color matches every time. Wood glue and sawdust.
Step 1: Mix, Putty, and Sand
So easy! Just put a few teaspoons of wood glue in a Dixie cup and start mixing in sawdust until you get a thick paste. When it starts to get just a little crumbly, you've got your filler.
Using a putty knife, push the mix into your screw hole, overfilling slightly.
Another method, and the one I prefer if I'm in a location where mess doesn't matter, is even easier. Squeeze glue over the area you need filled, dump sawdust on top, and knead in with your hands. Then brush away the excess sawdust/glue mix.
Give it a few hours to harden, then come back, sand flush, and viola! Holes plugged!
Happy gluing, and keep making awesome stuff!