Finding a Rainbow
No matter how much digital photography advice or photographic theory you get about rainbows, the first--and likely biggest--problem you'll encounter is simply finding one. If you're in rainbow hunting mode, it's not a bad idea to keep a camera in the car or otherwise carry one around with you.
That said, you can learn to anticipate prime rainbow conditions. To see one, you'll need both water in the air and bright sunlight. Rainbows commonly appear right before or after a storm hits your location, when it's raining nearby but the clouds have parted, allowing the sun to peek through. Rainbows always appear in the sky opposite the sun--so if the sun is in the west, look east for your rainbow.
Rainbows also have a very specific geometry. The top of the arch is always about 42 degrees above a line formed by the intersection of the sun and your head. That means you'll see more rainbows when the sun is relatively low in the sky; in fact, if the sun is higher than 42 degrees in the sky, the rainbow would be below the horizon entirely, and therefore impossible to see. Exception: If you're at a high elevation, such as on a mountain or on a cliff overlooking a valley, the rainbow might have enough room to form even when the sun is high overhead.
Make Your Own
I should point out that if Mother Nature isn't cooperating, you can create your own rainbow. Since a rainbow is just the right combination of bright sunlight and airborne water, you can generate your own with a sprinkler or garden hose from your backyard on a sunny day. Put the sun at your back and then spray water into the air in front of you. You'll want a fine mist or small droplets of water rather than a stream, so if you have a hose, plug it with your thumb to create a thin wall of high-pressure water. If you're lucky, you'll see a rainbow materialize. To photograph it, delegate hose duty to someone else.
Shooting Rainbows
This might be obvious, but I should point out that a rainbow is not a real physical thing; it is an optical illusion that is unique to your particular relationship to the sun and where you are actually standing at any given moment. You can't move closer to it; it will always appear to keep a constant distance from you. So while you can photograph a rainbow, remember that you can't lock your camera's focus on it, nor does the appearance of the rainbow affect the exposure setting.
That said, you can take advantage of the optical illusion to make a more interesting composition. If you move parallel to the rainbow (rather than getting closer or further away), the rainbow's relative position will change with respect to the background. That means you can make a rainbow photo more interesting by walking around until the rainbow's ends--where it intersects the ground--line up with something interesting. At the very least, by moving around you might be able to position the rainbow against a more interesting backdrop. In the photo on the right, for example, you can see a rainbow that's been expertly positioned to frame a tree.
Another element of the composition you can control is the zoom level. You'll need a fairly wide-angle zoom setting to capture the entire rainbow in a single photo, but by zooming in, you can focus the scene on just one part of the arc, as in the photo on the left.
Finally, you don't need to worry much about tweaking the exposure. You can shoot the scene normally with your camera set on its Program mode, but photographers will commonly underexpose the shot a little (you can set the exposure compensation dial to -1) to saturate the colors a little. Another way to saturate the colors: Use a polarizing filter. Experiment with your polarizer's setting, since using the polarizer at full strength can make the rainbow disappear from the photo.