Find an Honest and Knowledgeable Local Coin DealerIn order to make certain that you're consulting with an expert coin dealer, plus to ensure that the dealer you go to is an honest and ethical coin dealer, you should consider consulting the Professional Numismatists' Guild (PNG) directory first.
The PNG has very strict requirements for member dealers, which are outlined in the Five Tips to Find an Honest Coin Dealer. Mainly, you want to be sure you have recourse if something goes wrong, plus you want an expert dealer who has been reviewed by his peers for ethical behavior.
Another Source of Local Coin Dealers is the ANAAlthough I always recommend PNG Dealers before any others, because the PNG has by far the highest standards, in many cases there will be no PNG dealer local to you. In this case, you can check the American Numismatic Association's (ANA) Dealer Directory for someone.
The ANA also has a code of ethics, so you have some recourse if you get sold fake coins or unfairly graded coins, but the ANA is nowhere near as vigilant as the PNG in mediating disputes when people can't get a resolution directly. The up-side is that the ANA is far more likely to have a local coin dealer in your city.
Local Coin Dealers Can be Found at Shows and Coin ClubsIf you can't find a local coin dealer through the PNG or the ANA, check to see if there are any local coin clubs in your area. The ANA has a Coin Club Directory, plus you can try searching in Google for terms like "Your-City Coin Clubs" or "Your-City Coin Shows." Of course, substitute your city, or the biggest nearby city, in place of "Your-City" when you search. Many local coin clubs sponsor monthly shows, where several dozen dealers will set up tables and buy and sell coins. If you find a local club, but no evidence of a show, contact the club and ask them if there is a local show in town, or a dealer who be willing to discuss your coins with you.
Finding Local Coin Dealers Through Other SourcesIf none of the above methods are productive in finding a local coin dealer, the next place to check is the Yellow Pages, but not the online version! Get out the physical book and look at the ads in the "Coin Dealers" section. Many times, you can judge whether or not a dealer is a good choice simply by looking at his ad. Ads that say, "We buy junk jewelry and bullion" are not good choices. You want ads that say things like "Specializes in U.S. Coins" or in gold coins, or some kind of reference to the coins themselves, and not just the metal they are made from.
Last Resorts For Finding a Local Coin DealerIf none of the above methods produce anybody in your vicinity, expand your search to local newspapers, especially papers like the Recycler or Nifty Nickel. These are publications that come out weekly, and are full of mostly private party ads. They almost always have a Coins section where people will advertise to buy and sell coins on a private basis.
Try calling a couple of these folks and chatting them up. Some of them are part time dealer/collectors who sell on eBay or at occasional local shows, and they love helping out the novice or non-collector. But beware - some of these guys are pretty unethical, too!
If you must go this route, go prepared. Get yourself a copy of the Red Book and look up your coins yourself first, so you have an idea of what is most valuable, and don't let these guys "cherrypick" you. Sell them all or nothing, and never sell anything if it doesn't feel right to you!