I have a lot of non-Asian American friends who ask me how I make something, and when i tell them the ingredient came from an Asian food store they will reply with "I don't know how to shop there."
Sometimes they will venture in the store. They will do one of two things- head to the snack aisle where you can buy chips and cookies in interesting shapes, or buy overpriced exotic ingredients they will have no idea how to prepare. If the latter occurs, perhaps they asked a shopkeeper what to buy (remember they are salespeople and want you to spend money).
Since I live in the Chicago area, the Asian markets are huge and well priced. In more suburban areas you might need to head to a college town, which can be scary since you are sometimes the only one in the store with the owner watching you curiously. No one wants to look like they don't know what they are doing, so this is a basic guide to give you some good ideas of what to look for.
Of course, you can always buy "Asian" foods at Trader Joe's or Whole Foods, but they are not authentic and the prices are crazy.
WHAT TO BUY
1. Ramen: The Japanese and Korean brands are excellent. To reduce the oil, boil the noodles until they are halfway done, empty the water and pour about 1.5 cups water with favor packet to finish cooking.
2. Fresh Noodles: Go to the refrigerated section, and you will find noodles that are made fresh. They taste chewy and are 1000 times better than dry, most of the packages will have instructions in English for cooking. Some good ones to try are Yaki Soba, Udon, Soba, and Kal Guk Soo. If there is no favor packet included, you can cook them by just making a basic soup broth with vegetables. Many go bad quickly, so if you don't cook it within a week stick it in the freezer (unless it's udon or yaki soba. The Japanese have instant noodles which last forever.)
Not only do they taste better, most seem to be lower in calories compared to their dry counterparts. This is probably because they contain more water.
3. Asian Brand Tofu: Obvious choice. The firm has less calories than extra firm, and sometimes the tofu will be made right in the store.
4. Japanese Curry: This is an interesting product. It tastes sweeter than Indian curry, and will come in blocks which melt like chocolate as you are cooking. Add it to a stew of vegetables and potatoes and you've got a very good meal. They come in mild to flaming hot, most have cooking instructions in English.
5. Produce: They oftentimes will carry fruits which are not easily found in the American markets. If you are in a Japanese market they will be priced higher but better quality. This is the time you want to ask someone working at the store for help, they'll be able to tell you what just came in and tastes best. Persimmons, tangerines, and Asian pears are all good choices.
6. Rice: Buy a small bag of both brown and white rice, mix them together to cook.
7. Dumplings: These are usually in the frozen food section. Most Americans know them as wontons, and the Korean version is Mandu. It isn't hard to find the vegetarian ones these days. You can lightly fry them after thawing, or put them in a soup like you would for the refrigerated noodles.
8. Fresh fish: If you eat it, this is where to buy it. If it's a larger market they will have shrimp or other seafood that hasn't been frozen and needs to be cooked within one day. Canned dark tuna is also a lot tastier here.
9. Other items such as soup base, miso and spices are great. But do your research. Make sure you have a recipe in mind before buying so it doesn't go to waste. If something looks interesting, write it down and look it up later on the net.