How do I remove stickers and residue from a wooden door?
For this sort of residue, I highly recommend GooGone.

Mineral spirits, and Naptha will work, but the fumes they give off are much worse.  You should also do a test area first, in case the chemicals you're using causes problems with the original surface (removes it too, stains it, etc.)

You might be able to speed things up with a hair dryer (not a heat gun, it'll heat things too much), and a soft plastic scraper (something too hard will do more damage than it helps).  For the really stubborn stuff, GooGone + a plastic scrub pad (regular duty, not one of the heavy duty ones).

As for making it look better -- you might want to just strip the whole thing down and refinish it if it's particularly bad.