Most milk and many dairy-based products are gluten-free ... but, as always, there are exceptions.
Plain milk regardless of whether it's regular, skim or even heavy cream is gluten-free. Flavored milks, however, may not be safe, and you'll need to check ingredients to make sure. Malted milk products, including malted milkshakes, are not safe, since malt is made with barley.
Plain yogurt is safe, and I've had good luck with the Chobani and Fage brands. Many flavored yogurts but not all also are gluten-free. You'll need to check ingredients to be sure. Some yogurts come with cookies and granola, and you should avoid those.
The refrigerator case at the supermarket also carries eggs, which are gluten-free, butter, which is gluten-free, and margarine, most of which is gluten-free (always check the ingredients on margarine and shortening). You'll also find products such as Kozy Shack tapioca pudding, which is labeled gluten-free.
Some milk substitute products (such as soy milk and rice milk) are gluten-free, and some are not. Be particularly careful of gluten-free-labeled Rice Dream rice milk (found in the dry-goods section of the supermarket, not the dairy section), as it's processed with barley enzymes and many people report reacting to it.
Gluten-Free Cheese and Ice Cream
When purchasing cheese, most options should be safe. However, watch out for "beer-washed" cheeses, which seem to be a new fad among cheese makers. In addition, some manufacturers use wheat as a catalyst when making bleu cheese, so you'll need to contact the specific maker to determine if a particular bleu cheese is safe or not.
Lastly, beware of cheese that's been cut up and repackaged at the individual grocery store. In many cases, this repackaging takes place in the deli section on the same cutting boards where the staff makes sandwiches. I've been badly glutened by repackaged cheese. Look instead for cheese that was packaged at the manufacturer you may have to purchase more of it than you'd like, but cheese freezes well.
In the case of ice cream, beware of ice creams that contain chunks of cookies, dough or an unsafe candy. Check the ingredients and avoid anything with a gluten-sounding name like "Cookies and Cream" or "Chocolate Chip Cookie Dough" unless it's specifically labeled gluten-free. We've had decent luck with the premium ice cream products from Häagen-Dazs and Breyers.
Obviously, ice cream sandwiches are out unless you can find some that are specifically labeled gluten-free. But you can buy frozen fruit pops and other ice cream treats that are gluten-free for example, Dove Ice Cream Miniatures are a staple at our house.