You have too much friction somewhere -- the most likely culprit is the cable where it passes through housing. How old are the cables? Are the housings in good condition? Replace them with good quality cables and lined housing if you have any doubts -- these are not expensive parts and they are not expected to last forever.

Failing that, check the derailleur and shifters. If they're dirty clean them and lubricate them. If you still can't get it working take it to a shop.