How do I renovate this wooden furniture?
Can't tell what sort of varnish, if any, is on there, but it looks like it's come off in the worn places you care about anyway. Without a complete refinish job, you can try touching up much of the damage you picture with a swab that's been dipped in a dark oak (or red oak) stain. Just put a little stain in the places the finish is gone, wait 20 min. and wipe off gently. The result will not be as good as a complete strip, sand, restain and refinish job, but with the correct choice of stain color I've seen this method take things from fug-ugly to really not too shabby.
If you go this route, be sure to test it on a small, inconspicuous area first.