If it shifts to the gear when you hold the lever but immediately falls back down when you let go, then your levers are not operating and need to be flushed with solvent. Try squirting in lots of lube and working the levers up and down many times. If that doesn't work, take them off the bike and soak them in solvent for a day. That works more often than not, but occasionally the levers won't revive and have to be replaced. Squirt lube on all the pivot points of the derailleurs too in case they've become sticky.
If you can shift to bigger cogs OK but the chain doesn't want to fall back down to the smaller cogs, then replacing both cables is housing should do the trick.
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72