My advice for those not familiar with this system:
Lesson #1
Beginner's Gear Use.
1) DO NOT pedal backwards while changing gears - you may break something (that's for Sturmey Archer setups).
2) Keep the chain on the centre cog (controlled by the left gear lever) at the front most if not all of the time and just use all the gears at the back (right gear lever).
Ok, once you get used to that:
Lesson #2
Advanced Gear Use.
Now that you're used to all the gears at the back, you are probably finding that the highest gear isn't high enough (the one that lets you pedal slowly but go fast and is no good for going up steep hills). When you are on the HIGHEST gear at the back (smallest cog) use the LEFT gear lever to push the chain onto the largest ring at the front.
Similarly, when going up really steep hills with loads of shopping, you may find that the lowest gear (biggest cog at the back) is not low enough and you feel like you want to jump off and wheel your bike. This is the time to click the LEFT lever all the way back so the chain jumps down onto the 'granny gear' (smallest ring at the front)
ps. DO NOT go "big big small small" (eg. big ring at the front big at the back) because that will put the chain through a nasty S-bend.