I've been to a couple coin shows now, so I've combined my personal experiences with a summary of posts to rec.collecting.coins in response to this very question. For a listing of coin shows in your area, go to CoinShows where they are listed by state (US) and country.
Shows can be overwhelming. The vendor hall can be packed with row upon row of vendors and buyers. A little preparation will save you a lot of frustration.
Know what is considered a reasonable price for the coin(s).
Know how much you're willing to spend on each coin.
Know the correct name and description of the coin(s) you want.
Know the desirable features of the coin(s) you want (see all 4 legs of the buffalo,
x number of tail feathers on the eagle for Morgan dollars, etc).
Having these things isn't essential, but it will help you keep track of who's offering the best prices and where to find them again once you've finished touring the vendor hall. These things can also help you buy just what you're looking for and stay within your budget.
Bring cash since most dealers will not accept personal checks or credit cards. There may be ATMs available at the conventional hall or wherever the event is held, but they may charge high fees or even run out of cash.
Bring a list of exactly what coins you need, plus a list of coins that you'd like
but that are your second choice, in case your first choice coins aren't available. It's very easy to impulse-buy and get something you already have, or forget something you wanted.
Bring a magnifying glass to help you determine the differences between two examples of
the same coin and of similar grade. A vendor may provide a lamp or two, since the venue's lighting won't usually be bright enough, but not all of them provide magnifiers.
Bring a notebook to keep track of who offers what coins at what prices. Write down the vendor's name, and their location within the dealer's hall. Most large halls are labelled with row numbers or table numbers.
Remember, you're not obligated to buy anything! I know it's easy to feel guilty if you spend a long time poring over coins only to end up not buy anything. As long as you're not blocking other people from looking, it's expected that you'll spend time browsing. If the vendor has answered questions or been especially helpful, thank them before moving on, and ask for a catalog, business card, or brochure.
Shop around: look through the entire bourse (dealer area) to see who offers what coins at
what price and what quality, then go back to view the coins that look promising or the vendors that had what you wanted. Nothing is more discouraging than to buy a coin at the front of the hall, only to find a better quality example for a better price at the end of the hall!
Eye appeal and the interaction you have with the dealer can determine what coin you buy as much as the price and the quality.
You might not want to buy all your coins through one dealer. The convenience is nice, but it's good to establish a rapport with several dealers, especially if they're local to you.
Ask if the dealer has a business card, a website, mailing lists, and price lists for future reference. It's very easy to forget a good dealer's name among the hundreds listed in the conference program. This is also a nice gesture to thank them for their time if they've been helpful or if you browsed a long time without buying; it shows that you may buy from them in the future, so it wasn't a waste of their time.
If it's not too busy, take the opportunity to learn about grading and cleaned vs natural coins:
take at least 2 or 3 examples of the same coin and note the differences between them, especially if there is a substantial difference in price between. Why is one worth more than the other?
Don't drag a spouse/child/friend along who isn't interested.
Wait for a lull to ask questions.
Ask for permission before handling coins, especially if they are not in some sort of protective holder or if they are in a display case.
Dealer tables can get crowded. If you're just browsing, make room for those who are buying.
I would recommend not buying anything the very first time you go to a show, especially if it's a recurring one in your area. Just get a feel for how they're set up, what is offered, and how much the items you're interested in cost. This way, it takes the pressure off you. Even if you don't come home with anything, that doesn't mean it wasn't a successful trip. Whether it's finding that missing piece for your collection, scoping out market prices, learning more about grading and the latest trends, or just interacting with fellow collectors, the main point of going to a coin show is to enjoy yourself!
Thanks to Lee Cherry, Bill Chin, Marada C. Shradrakaii, and "Mbagby" for their responses.