Food To Avoid
It's not coincidence that "foods to avoid" is listed prior to "foods you can eat"! In my experience, I've had much more trouble knowing what NOT to eat. There's no worse feeling than sitting in a restaurant, looking at a menu, and just having no idea. Likewise, while the list of foods you CAN eat is rather infinite, the list of foods you CAN'T eat is pretty manageable, and thus is an easier starting point.
These lists are NOT exhaustive! This is meant as a resource to get you started. The most important thing to do is to look at ingredient lists on product labels in the grocery store. Soon you'll get a feel for what you can and can't eat, and will be able to ask simpler question in restaurants. Asking a waitress if something contains soy sauce is easier and less awkward than asking if it contains wheat!
The obvious ones
Bread: Unless it's advertised as wheat-free, you should assume it has wheat in it. Even if the grain is listed as something other than wheat (such as rye or oat), these breads almost always also use wheat. (The one exception is some old-fashioned rye breads.) Take a look at the ingredient lists on the bread aisle in the supermarket, and you'll see what I mean.
Bread-like products: Pita, naan, wraps, bagels--they're all made from the same ingredients as bread (except, perhaps, yeast). You're ingenuity is commendable, but you'll have to find something else to eat your sandwich one! And don't even ask about breadcrumbs and croutons--you can guess where they come from!
Pasta: Words like "durum", "semolina" and "macaroni" all mean the same thing--wheat! Standard Italian-style pastas simply aren't going to cut it. The good news is that, when dining at home, there are alternatives.
The not-so-obvious ones
Soy sauce: My #1 offender! This took a while to discover, because one wouldn't think that soy sauce has wheat in it, but it's right there on the ingredient list. Unfortunately, this means that anything at a Chinese restaurants (except something with no sauce or a white sauce) and most items at a Japanese restaurant (teriyaki's top ingredient is soy sauce) are now off-limits. But don't freak out just yet--there's a great substitute.
Barbecue sauces and marinades: Unfortunately, wheat is frequently used as a thickener in sauces of all sorts, especially in store-bought stuff and in chain restaurants. Before ordering anything in a restaurant that comes with a sauce, be sure to ask if it contains wheat; in my experience, they'll be prepared for this question.
Salad dressings: Many of them, even if they're not Asian-style, contain soy sauce since, let's face it, it tastes good. Usually white dressings are a safe bet, but they are obviously not a healthy choice either. I'll never buy a salad dressing without inspecting the ingredient list.
Cereal: Even if it's not wheat-based, many cereals contain wheat ingredients.
What to look for
More and more companies are including a "CONTAINS WHEAT INGREDIENTS" statement under their ingredients list, but as it's not required, you can't depend on its absence to mean the opposite. Here's a quick guide to what various labeling practices mean:
CONTAINS WHEAT INGREDIENTS: Obviously, this is off-limits. Move on.
DOES NOT CONTAIN WHEAT INGREDIENTS: Score! We have a winner.
MADE ON EQUIPMENT SHARED WITH WHEAT: If you have a wheat allergy, you may want to avoid this product, depending on your level of sensitivity; consult your doctor. In my experience with wheat intolerance, this is fine--trace amounts are extremely unlikely to upset your system.
MADE IN A FACTORY THAT PROCESSES WHEAT: In all likelihood, there will not even be a trace amount of wheat; the food processor is doing this just in case, to cover their you-know-what. Unless you are an anaphylactic risk, you should be fine eating this.
DOES NOT CONTAIN GLUTEN: More information than we need, but wonderful nonetheless. Wheat contains gluten, so by power of deduction, if there's no gluten... there must also be no wheat!
CONTAINS GLUTEN: This isn't too helpful, because while gluten-free means wheat-free, the opposite isn't always true. In other words, gluten can come from sources other than wheat (such as barley, rye, spelt, etc.). You will still need to inspect the ingredient list to try to determine the source of the gluten--if it's not wheat, then you're in luck.