Hi DSR, Cozumel has some great beaches. Chankanaab is my favorite because of the snorkeling. Paradise is very similar to Chankanaab, but doesn't have good snorkeling or a zoo or as many shops, but does have a nice beach with chairs, restaurant/bar, and water toys. Nachi Cocom is very nice, but you must have reservations, they only allow 100 guests. They have a swim up bar, restaurant, and a nice beach area. (We stopped there to get out of the rain, so didn't check out the snorkeling). There's a Carlos 'n Charlies, and many more on the way down to the southern end. Chankanaab and Paradise will be the least expensive to get to via cab, Nachi too. There's plenty of shopping at either port there, too.
Grand Cayman my favorite beach is 7 mile. There have been some good posts for Rum Point, but we haven't been there, yet. The snorkeling is really good in Grand Cayman. There's a dive shop (Abanks) that used to let you go into the water there, plenty of wrecks right by the tender area as well as coral and sea life. Stingray City is the main attraction there, which we've done a couple of times. I'm just as happy taking a bus done to 7 mile and spending time there or snorkeling. It's a shopper's haven there, too.
Have a wonderful time!