What I'd do:
1) Release the cable from the derailleur.
2) Visually inspect cable, housing, and derailleur for anything that looks like an issue. Frayed or kinked cable or housing, anything gumming up the derailleur free movement, anything.
3) Manually move cable forwards and backwards through all housing to make sure it slides smoothly.
4) If it doesn't slide smoothly, replace housing sections and/or cable.
5) Dial all tension adjusters in (as if you're letting off tension), then back out a turn.
6) Reattach cable to derailleur.
7) Go here:
http://www.parktool.com/blog/repair-help/rear-derailler-adjustments-derailleur to make sure your limits, b screw, and tension are adjusted correctly.
8) Those steps almost always solve my shifting issues. If they don't my next step is to check derailleur hanger alignment. You probably don't have the tools for that though.
Edit: Oh, and make sure your rear wheel is correctly in the dropouts and tightened down. Sometimes a misalignment of the wheel is all that's wrong.