Okay, it is time for the adjustment of the rear derailleur. First of all, much has been written on this subject already. In fact, you can find an article on the adjustment of a rear derailleur here and here from this site. I suggest you read all you can get, and take what you can understand from all of them. I may not explain something like someone else does that clicks with you. With that in mind, lets take a look at a typical derailleur.
The barrel adjuster is located where the cable exits the rear of the derailluer housing
The main adjusting feature on your derailleur is the barrel adjuster. It usually is located at the exit point for the rear cable housing from the deralleur body. SRAM derailleurs don’t use this arrangement on the mtb side, and niether do the new Shadow derailleurs from Shimano. The barrel adjusters for these systems are located at the shifter. At any rate, they function the same way. If your derailleur cable is too slack, you turn the barrel adjuster counter-lockwise to take out the slack in the cable. If the cable is too tight, the adjuster is turned clockwise to loosen the cable. Now might be a great time to lube the derailleur cables too. Check out how to do it here.
Set ‘em and forget ‘em!
The two screws found on most derailleur bodies and usually marked “L” and “H” are the limit screws. They keep your rear derailleur from traveling too far inwards or outwards. Once these have been set, they generally do not ever need to be fiddled with again. However, it is worth checking the limits of travel on the derailluer just in case. “L” stands for “Low” and is the screw that limits the travel of the derailleur to the lowest gear. “H” is “High” and limits the travel to the smallest cog, or outside of the bike. Clockwise turning of either screw will limit travel. Counter-clockwise turning of either screw will allow more travel.
Manually checking the limits of derailleur travel.
To check the limits follow the instructions here. Once you have done this, then adjusting the rear derailluer can effectively take place. If you didn’t lube the cables, go back and do that. Now, if you pedal and shift the bike’s gearing system, you should see immediate results when initiating a shift. If the chain hesitates when you shift from a small gear to the next larger, then you may want to add tension to the cable. (Turn the barrel adjuster counter-clockwise a 16th to an 8th turn.) Check it again. If it still hesitates, add more tension to the cable. Repeat the process until you get the shifting to hesitate on the way from a bigger cog to a smaller. Then back off the adjuster an 8th of a turn. If the system is doing the opposite from the get go, reverse the procedures until satisfactory shifting occurs.
Note that it may take some trial and error your first few times to get it right. Also, if your derailleur is not hanging perfectly parallel to the cogset, your derailluer hangar may be bent, or your derailluer damaged. That would mean it is time to see the experts at your shop. Also, if your cable housing is cracked or kinked, or your cables frayed, it might behoove you to replace all of that before you move on.
Next: Front derailluer adjustments.