How to Adjust a rear derailleur
Use a 5mm Allen wrench and loosen the bolt holding the cable on the rear derailleur. Thread the shift cable through the cable stays and tighten it in the rear derailleur with as little slack as possible. Make sure that the adjustment screw on the shifter is in the mid point of the adjustment.
Shift the chain into the big chain ring and the smallest cog. As you turn the cranks to pedal the bike, make sure that the chain rides silently on the small cog. If it does not and tries to skip up to the next biggest cog, unscrew the high (marked H) setting screw on the back of the derailleur. If the chain is trying to skip out beyond the small cog, tighten the screw slightly.
Perform adjustments in 1/4 turn increments. Next, shift the chain into the small chain ring and biggest cog. Perform the same check on the cog. Loosen the low (marked L) screw to move the chain to the inside and tight to move towards the outside. After the screws are set, shift to the middle chainring and the smallest cog. Check to see if the shifters will shift the chain into the next highest cog when shifted once.
If the chain does not shift, unscrew the cable adjustment screw on the back of the derailleur until the cable jumps up onto the next cog. Try shifting back and forth between these two gears. If the cable does not want to shift up quickly enough, increase cable tension by unscrewing the cable adjustment screw. If it does not want to shift down to the smallest cog, decrease cable tension by screwing the cable screw in.
Repeat this step for the next couple of gears until all are shifting correctly.