You have several options:
Coin dealers are specialists in the coin market, but they are out to make a profit so they may not give you the best possible price. You can find dealers in your area in the phone book (try both "collectibles" and "coin dealers") and through online dealer directories (see the links section).
Coin clubs are another good place to find people who are looking to buy coins and meet fellow collectors. In the US, you can try the ANA Coin Club Directory. You might want to do a general net search for coin clubs in your area, too, in case they aren't listed in the ANA directory.
Coin shows are fun to visit even when you're not trying to sell coins. They're convenient when you are trying to sell them since there are many dealers all assembled in one place. The site Coinshows lists coin shows in the area by state.
You still have some options! Read on...
Antiques shops and consignment shops may accept coins; pawn shops will almost always accept them. Almost every town has at least one out of three. Antiques shops will usually just buy the coin outright from you; with pawn and consignment shops, you won't receive any money until the item actually sells. In the case of all three, you may not get the best price for your coins since they don't specialize in coins. (I suppose there's also the possibility that they might overestimate the coin's value if they don't know the market, but I wouldn't count on that.) Still, it's a convenient way to get rid of coins quickly and get at least a little money for them.
There are two Usenet groups dedicated to all aspects of coin and banknote collecting:
If you don't have a newsreader, go to Google Groups and subscribe so you can read and post to the newsgroups. As a courtesy, put FS (For Sale) in the subject line of your post, then in the body of the message give as detailed a description as possible of the coins or banknotes that you're selling (country, year, denomination, and approximate condition) and a rough estimate of the price you're offering. If you have a scan of the coin or banknote, Do not post scans to the group! It is text-only; images clog it up. Offer to send the image as an attachment through email on request or post a link to the image if you have it on a server somewhere. Check back each day for about a week for responses to your post.
Online auctions are not a method I'd recommend to a beginning coin collector. It takes a lot of preparation and research to run a successful online auction. It's generally only worth the effort for large amounts of coins or coins of fairly high value. Perhaps in the future I'll put a separate FAQ sheet in on how to run an online auction, but for now I wouldn't recommend them to a newbie.