Pick a store that you know has good standing and solid reputation. Gourmet cheese is a perishable, highly delicate product, and you should be able to trust your source completely. Quality is of utmost importance.
Make sure the costumer service or product assistants are knowledgeable, helpful, and available to answer your questions. It's their job to help you out and advise, after all, they are the experts.
The more variety in the store, the better. Not only because you get better selection, but it typically means the store get requests for highly sophisticated cheeses frequently.
When buying online, make sure you get the storing, and handling instructions. Not all cheeses should be stored in the same manner, and unless you're an expert, it is the store's task to provide this information to you.
Don't let your eyes be bigger than your stomach. Cheese should be used and eaten quickly, as it tends to go bad pretty easily (pressed or cooked cheeses, as well as well-ripened goat cheeses last longer).
Look for fermier or artisan cheeses, which are usually the best. Also look for DOP or AOC labels. The most of those gourmet cheeses you find, the more high-end the store will be.
If you're looking for gourmet cheese gifts or gourmet cheese gift baskets, cheese assortments, cheese boards, etc, make sure the selection comes with an explanation and description of the cheeses selected.
American cheese stores usually don't carry American cheese. There are a few good US-made cheeses, like Vermont cheddar, but typically a good store will carry mostly French, Italian, Spanish, Portuguese and Dutch cheese (amongst others).