You likely have a setup similar to this (for a Trek FX2):
Front 3 chain rings
28/38/48 tooth chain rings
Back 8 cogs
11-32

It's best to avoid the extreme cross-chained combinations. For example, the 28 chain ring and the 11 cog. But an occasional cross chain won't cause any problems or wear out your drive train.

On the small 28 chainring, the larger 6 cogs are fine, and avoid the two smallest cogs.
On the middle 38 chainring, use any of the cogs.
On the large, 48 chainring, the smaller 6 cogs are fine, and avoid the two largest cogs.

Many riders use the middle chain ring on flatter roads, the small chain ring on hills, and the big chain ring rarely--often just on downhills.

Cadence -- pedal rotations per minute, in rpm. (To get a feel for this, on a quiet road, count the right side pedal strokes for 20 seconds, then multiply by 3.)
A good goal is to be "spinning" instead of "mashing" the pedals.
Mashing is pushing hard on each pedal stroke. It's best to save that for really steep hills.
Spinning is a fairly fast cadence, with light pressure on the pedals. It's better for riding longer rides. It uses more of your cardiovascular system, not just your leg muscles. You might try clicking one gear easier than you normally would use, and get used to that faster pedal cadence.

The chart
Here's your gears at some typical spinning cadences. the big 48 chainring is in blue, the middle 38 chain ring is in black, and the small chainring in red.

There's lots of overlap between the 3 chain rings. At 15 mph, for instance, the small chain ring is near the top end of it's range, close to being cross chained small-small. The middle chain ring can go down below 10 mph, or above 20 mph. The big chain ring is close to being cross chained big-big.

So, in general, try out using the middle chain ring, and shift to the small before you reach the base of a hill. Or if you are riding slower that day, you'll use the small chain ring most often.