Most modern road bikes have two chainrings in the front, and a 9, 10, or 11-speed cassette in the rear. These are your gears, and they allow you to climb more easily, go faster downhill, and ride almost any terrain.
The left shifter is for the front derailleur and the front chainrings, which are your biggest changes in gear. The right shifter is for your rear derailleur and rear cogs (or cassettes), which are your smaller changes in gear. Left=Front=Big. Right=Rear=Small.
Start in a gear that's appropriate for your terrain. In general:
Once you're going, use your rear shifter to select and fine-tune your cadence. Your cycling cadence is the number of times per minute you rotate the pedals (measured in RPM's), and is the secret to riding faster for longer periods of time.
Understanding cadence is key to making good decisions with shifting. It is a calculation based on how you're feeling, the slope of the road, the wind conditions, the group you're with, the intensity of the ride, your fitness level, and a variety of other factors.
Once you know how it all works together, the right combination of shifting and cadence allows you to minimize muscular fatigue, build aerobic endurance, and ultimately go faster with less effort. When coaches say "Spin to win," this is what they're talking about.
A good general guideline for cadence is 85-100 RPM. This allows you to use less force per pedal stroke and helps you easily adjust to changing conditions. When climbing hills, it can go a little lower, around 70-85 RPM depending on the length and grade of the terrain.
As you ride, notice how small changes in your rear shifting affects how fast or slow you are pedaling. When you're cycling in a gear too big, you end up pushing harder and your cadence slows down. If the gear is too easy, you spin out. Your optimal cadence is the balance between these two scenarios. You want enough resistance to maintain your speed, but in an easy enough gear that your legs spin efficiently.