Arright, here's the skinny from a guy on Year 2 of mountain biking.
The left lever switches your chain ring, the gears on your pedals.
The other one is the gears on your cassette, which is on the back tire.
Basically you want to be on your middle chain ring, #2, most of the time. For hard uphill you would use #1 (small chainring), and for long flat distances you would use the big one, #3, to get more wheel rotations out of each pedal stroke.
Your gears, the right lever, are more akin to gears on a car and should be relatively easy to figure out...but it's different on every bike. But the general rule is you want smaller gears for steeper terrain because you get more torque. If I go down to my lowest possible gear ratio on my bike and pedal from a stop, my front tire tries to come off the ground and I'll dig holes if the soil is loose enough.
GENERAL RULES TO NOT THROW YOUR CHAIN OR WORSE:
Do NOT go to the extremes of gear ratio. Your middle chain ring is the only ring that will let you use your full range of gears. If you look down at your chain and it's at a sharp angle or you can hear it rubbing/clicking, you need to shift back to a closer gear.
Also, you'll want to try and get used to dropping down a gear or two before you reach an uphill. This will not only save wear and tear on your cassette, but will make your ride smoother and let you keep speed as you go hit the incline.
For average distances of flat ground, I'll stay in my middle chain ring and just cycle up through gears until I max out or have to go uphill. For nothing but flat riding (rare for me), I'll start in the middle chain ring on 1st gear, then work up to 2 or 3 and then switch to my larger chain ring and adjust appropriately.
Again, this might be different for you as it depends on you, your bike, and your environment. I ride trails way more than I do anything else and my bike it built for that purpose.
