1. Please buy a swimsuit that fits your child correctly (and is made for swimming, not looking cute). This means a snug-fitting, one-piece suit (a suit with raceback straps like this one is awesome!). Two-piece suits are adorable but the absolute worst. One-piece suits with flouncy skirts (like this one) are also cute, but they make it hard for the child to see what their legs are doing. It's the most frustrating thing in the world to try to teach a child to swim who must constantly adjust the straps, waist, etc (we often ask each other, "Can't the parents, who are watching from the stands, see what a hassle it is?!"). I know it's tempting to get the next size up to try to save a little money (kids grow fast!), but it's not worth the wasted time.
Also, if you intend to have your child in swim lessons (or any other water-sport) for awhile, you need more than one suit, I promise. What happens when it's January, there are no suits in the stores, and your child's swimsuit suddenly falls apart (it happens) or is misplaced? A water aerobics teacher told me once that the rule of thumb for swimsuits is that if you're in the water more than twice a week, you'll probably need a new suit every 12 weeks or so.
If your child frequently complains about being cold, consider getting them a rashguard (Lands End sells lovely ones for both boys and girls, but even the cheapest neoprene rashguards will help immensely).
2. If your child's hair is long enough to reach their eyes when wet, tie it back before sending them to class. (Little girls AND little boys.) Better yet, get them a swim cap (and learn how to put it on: have the child hold the "front edge" of the cap with both hands very near their forehead; count "one, two, three!" and stretch the "back edge" of the cap back toward the nape of their neck. Tuck any loose hairs into place. If you have a little girl, put her hair back in a bun at the back of her head before putting on the cap).
I taught a little girl over the summer whose thick hair - no kidding - reached below her waist. The parents never pulled it back, so after trying for several lessons to get her to blow bubbles, and having her panic every time her hair covered her nose and mouth, I got into the habit of braiding her hair before taking her into the pool. (I did often wonder why her parents, who could clearly see her lessons, never bothered with this step themselves lol.)
3. Don't buy cheap goggles. The best goggles don't have Disney or Marvel characters on them. ;) Make sure they're sized for your child's age (child, youth, etc) and pick ones with a solid nosepiece (like these) and not an adjustable one (like these). (By the way, that first pair is my absolute favorite kind to put on little swimmers, and they're the ones I snap up on clearance to stock the swim bag with!) A pair with divided straps (like the first) is better than one without. If you have a very young child, invest in a pair of Frogglez. You can also try a hybrid swim mask if they have trouble getting a good fit - these cover a little more surface area so sometimes they work better for small faces - but DON'T buy a snorkeling mask that covers the nose. The first and most important skill when learning to swim is learning to control your breath, which is impossible to do if your nose is covered.
Teach your child to hold the goggles over their eyes while you (or their coach, etc) adjust the straps. There's less chance of injury this way, and you'll get a better fit. After they're done swimming, gently rub a little lotion, hair conditioner, etc over the inside of the lens, and swish it off in the water (don't rub!) before the next swim to maintain an anti-fog coating.
4. Don't send your child to class wearing jewelry, a watch, or, heaven forbid, toting a toy. It's hard to get small children to focus in the best conditions. At best, your child will be distracted, at worst the coach will spend precious class time fishing a lost charm out of the filter amid tears. If the coach wants to use toys to help your child learn to swim, they will provide some. Otherwise, send your child to lessons without accessories!