The biggest problem with many amateur guitarists is poor tone, and that's the first thing you might want to correct before you start placing a microphone for recording or live sound reinforcement. Take a listen to your amp and guitar combination on microphone-level -- that is, where the microphone would be placed when recording. Adjust your tone so that you're happy with it, but remember one thing: the low-end will be increased with a microphone placed close to the source, something called proximity effect.
Selecting The MicrophoneYour preferred microphone for this will be a dynamic microphone for it's ability to withstand high volumes, and it's relatively low price. Sure, there are great condenser microphones that sound wonderful on guitar amps -- I personally love using a combination of dynamic and condenser, or dynamic and ribbon. But, with most budgets, a single dynamic microphone is a great choice.
The universally-accepted choice is the Shure SM57 ($99), but the Sennheiser E609 is also a popular choice, and is my preferred microphone for guitar amps. My favorite condenser microphone for guitar amps is the AKG Solid Tube, but depending on your preferences and needs you may find something different works best.
Placing The MicThe first thing to understand in microphone placement on guitar amps is that the sound doesn't resonate from the center of the cone; it comes from the off-axis - or "off center" area. How you position the microphone is really up to you and what you want your recording to sound like. If you're wanting a louder, more treble-filled sound, you want to place your microphone towards the center cone. If you want more of a warm tone, start moving out towards the center.
You might also try using two microphones on your amp - one close up and one farther back. You also need to remember, when running your amp in a recording situation, recording it at the highest volume possible while still achieving your desired "tone" is really recommended.
It's also not unheard of to mic the back of a guitar amp -- sometimes, blending in the tone from the back of the amp is a great way to get the full picture of an amp's tone.
Once you record your track, you may also find that "doubling" the rhythm track makes it sound way better than simply panning it in the center. Record the rhythm track, pan it hard left or right, and record it again, panning that to the opposite side -- even the steadiest hand will have some variation in the playing rhythm, and it'll make a much fuller, deeper sound on your mix!