The left hand control moves the chain on the front gears (by the pedals; these are called "chain rings"). The right hand control moves the chain on the back gears (on the rear hub; these are called "cogs").
You can only shift while turning the pedals. Shifting simply "pushes" the chain sideways; it needs to be moving in order to actually jump to the adjacent ring or cog.
Find some level ground. Put the left shifter into the middle (2) position (lift the back wheel and turn the pedals if necessary to move the chain). The chain should now be on the middle chain ring in front. Put the right shifter into the (1) position (again, turn the pedals). That should move the chain to the largest (leftmost) cog in the back.
Now start riding. When you're moving smoothly, move the right shifter to the (2) position. The chain will move to the second-to-largest cog. You should be going a little faster now. This is just like shifting from first gear to second gear in a car. On level ground, you can probably continue shifting to 5 or 6. By that time, you will be moving much faster, and it will be increasingly hard to pedal.
If you start to go up a hill, it will quickly get hard to pedal. Now you need to downshift, so click the right side back a position or two toward (1). If the hill is really steep, you can get even lower gears by moving the LEFT shifter to the (3) position. That will move the chain to the smallest (leftmost) chain ring.
The key is that you have to shift to respond to changes in terrain (and winds). You have to practice, with the goal to be able to keep your pedals turning at a fairly rapid and constant pace (called "cadence").
For starting out, though, don't worry too much about it. Just have fun and concentrate on riding safely. When you go up hill, shift the right side to lower numbers; when it gets easy to pedal, shift the right side to higher numbers. When you are comfortable with the right side, you can start playing with the left side to see what that does for you...