You have three rings in the front, 8 or 9 in the back.
When you are in the biggest chain ring in the front, and the biggest gears in the back, those are your "lowest" gears: you use those for going up hills or going slow. The very lowest is your "granny" gear. If you are new to biking I bet you will be using it a lot ;-)
Using the biggest chain ring in the front is when you want the most speed and power... going fast down a grade for example.
The middle section of your casette ( what we call the back cluster of gears) are your most useful.
When you are in one gear and the pedaling gets easier, you want to go "up" a gear, up a number on the back, into a smaller ring on the back. Or, you can go up (bigger) in the front lets say from the inner to the middle chainring.
Same thing applies in reverse. If your pedaling is getting harder ( going up a hill) you want to go "down" a gear in the back... onto a bigger gear in the back, down a number.
And, go down one in the front,
Do's and Don'ts
Do not climb in a "big" or "high" gear. This is really hard on your knees. You want to be pedaling easy, not pushing.
Do not cross chain. This is, using your smallest ring in the front with the smallest gears in teh back. Or, big chainring in the front with biggest gears in the back. This is really hard on your chain. If this is where you need to be gearwise... go to the middle chain ring in the front and then find your correct gear in the back.
Do not shift under load. This means, with a lot of pressure on the pedals. Try and set up for what gear you need before you need it ( eespecially when going up hills) or, take the pressure off the pedals. Shifting underload is a major cause of chain suck.
hth
~Irulan