You can first visit your county elections website since they offer an easy registration lookup tool, which lets you to confirm if you are registered. You can also call your county elections department if you desire the reassurance of human contact. Find out more details on http://www.ehow.com/how_8527613_confirm-registration-vote.html.
It varies by state, but it is very common to go through your MVA or DMV. It's done that way because 1 of the first things that you're supposed to do (30 days) is to get a new drivers license, even if your in the same state you're supposed to notify them & either get a new license or get a change of address card that you can carry with you. Someone in the area can tell tell you where to go & who to speak to. It's normally going to be the 2 mentioned or the library, post office, local schools, comm. college etc.
If you voted in the last election, you should be automatically registered to vote in the county where you last voted. However, if you have moved, changed any of your demographic data, or changed your party affiliation, you can reregister. See your local voting administration.