I'd start by shifting to about the middle of the cassette and eye-balling the chain on its sprocket, from the rear of the bike. Is the derailleur's jockey wheel exactly centered on the sprocket? It really should be. There's some tolerance allowed, but you really do want it centered there. Turn the barrel adjuster until it is -- CCW to move the derailleur in towards the bike (pulling more cable, essentially), and CW to move the derailleur out from the bike (loosening the cable). It sounds like you're having trouble with the derailleur moving enough to get a good downshift (to a larger sprocket), so it sounds like the derailleur may not be centered under each sprocket. Of course, there could be a variety of other issues, but this is an easy place to start.
You should, at the very least, be able to influence shifting quality somehow with that barrel adjuster. If it seems like you can turn it and turn it and nothing changes, then something else may be going on.