This is a list of some items that may not be on your shelves, but that I find make gluten- and dairy-free cooking and eating easier. There are obviously no wheat or wheat flour, oatmeal, and barley in the book and certainly no milk foods (lactose).
I assume that most of us have salt and pepper and other common seasonings and ingredients. Here I list only those items that may be new to the cook. It used to be hard to find many of them and it often meant a trip to a specialty shop. Many can be found in ordinary markets or online.
GRAINS: Amaranth, buckwheat, chestnut flour, corn (flour, masa harina, meal, polenta, starch, popcorn), garbanzo flour, potato starch, quinoa, rice bran, rice flour, tapioca flour, and teff (seeds and flour). See the entries in Stiff Upper Lip: The Starches (page 210) for more information.
PASTA, NOODLES, ETC.: Gluten-free pasta, rice (paper, wrapper, noodles, sticks), and mung bean noodles/threads. Gluten-free pasta can be bought and is satisfactory (see the chart on page 39). Rice and mung bean products are excellent gluten-free substitutes.
OILS, VINEGARS, AND CONDIMENTS: Harissa (a Moroccan spicy seasoning paste), rice vinegar (a mild vinegar less dominant than others), safflower oil (neutral in flavor and can be heated to very high heat), toasted sesame oil (chestnut brown in color, it has a warm, delicious flavor; it should not be confused with golden-colored plain sesame oil and it should not be overheated), sherry vinegar (brown in color, somewhat unusual and rich in flavor), and gluten-free soy sauce.
COCONUT MILK: While coconut milk has a flavor of its own, it can be substituted for regular milk in many recipes with appropriate seasonings. Take care; there are thick coconut milks. I use the ordinary kind, buying it in cans.
SEASONINGS: Anchovies (whole packed in oil and anchovy paste), black mustard seeds, caraway seeds, capers (packed in salt), chocolate (dark—no milk chocolate), espresso powder, fermented black beans (available at Asian grocery stores, these have a salty deep taste), dried mushrooms (shiitake; porcini, which give depth of flavor to ordinary mushrooms), ras el hanout (a spicy Arabic seasoning powder), sardines (whole packed in oil), star anise (an Asian spice with a faint licorice taste), and sumac (used in the Middle East, deep dark purple sumac grows on bushes).
Keep in mind that the more usual seasonings, seeds, and herbs (fresh and dried) can vary in flavor, often according to their place of origin. For instance, Syrian oregano is more pungent than Italian—or, for that matter, than what I grow myself. Spices also diminish in intensity with time.