I agree it sounds like some maintenance would be more cost effective than a new bike, if you like doing that sort of stuff. You might want to purchase the appropriate tools to get into the hubs and replenish the grease there, and probably get the cones tightened correctly.
Fixing shifting, Andy_K suggests that you "might" need new cables and of course he's right, but if you want to save some time and be done with it upgrade that "might" and get new cables and compressionless cable housing at the start and go from there.