There are three adjustment points on your derailleur - a barrel adjuster, and H/L screws on your rear derailleur. Think of H as 'hard gear' and L as 'lazy gear'. The barrel adjuster is for fine tuning cable tension.
If the derailleur won't shift all the way down to the smallest gear, it can be one of two things:
* Incorrect H screw adjustment - To check this, unbolt the cable to your derailleur and pedal a few times. The chain should drop to the lowest gear. If it doesn't drop to the lowest gear, then the H screw is too tight and must be loosened until the chain drops to the lowest gear.
* Incorrect cable tension - Once you know that your H screw is set properly, just pull the slack out of the shifter cable and bolt it back on.
If the derailleur won't shift all the way up to the biggest gear, you need to adjust the L screw (that's the cog you use when you're feeling lazy). Put the bike in the easiest gear it will go to. Loosening the L screw and shifting up/down will allow the chain to travel further towards the spokes, tightening it will move the chain the other direction.
After you get the H and L screws set properly (so the chain can go all the way up and down the cassette) you'll probably notice a few gears that don't shift nicely. No problem. Put the chain on the second smallest gear and shift to the third smallest gear. If this is slow to shift up, loosen the barrel adjuster a notch. Then switch from the third smallest gear to the second smallest gear. If this is slow to shift down, tighten the barrel adjuster. Keep playing with it until you're switching seamlessly from 2-3 and 3-2.
Then you should be good.