Welcome aboard, Denise! Setting sail on the Disney Fantasy is going to be a very magical vacation for your family.
I always recommend choosing one of Disney Cruise Line's Port Adventures versus going off in the port on your own. Disney takes care of all of the details and ensures that you are back to the ship in plenty of time. I promise that Disney's adventures are worth every penny versus trying to head out on your own. But if you do decide to explore the port on your own, I have a few suggestions.
Grand Cayman has some beautiful beaches along Seven Mile Beach. I've heard great things about some guests taking a bus for about $5 per person each way to Surfside Beach. There are umbrellas, chairs, snorkeling gear and other water activities available, all at an additional charge. The last time I sailed, a friend went there and told me that umbrellas started at $4, and chairs cost $5 or $6 each to rent.
You won't find any beaches within a short walking distance from the cruise port in Cozumel, so you'll need to take a taxi. Playa Palancar is supposed to be a nice spot accessible via a short taxi ride. But, I very much prefer
Playa Mia Beach Club which you can enjoy with Disney's Port Adventure. Your daughter will love the waterpark area there and you'll enjoy relaxing in a lounger on the beautiful sandy beach while you sip a fruity drink!
Costa Maya's most popular beach on which to relax is Mahahual, which is a just a short taxi ride from the cruise port area. It's a cute little village and there's quite a bit available there at the beach, making it a nice place to spend the day. Look it up online, it's beautiful!
Enjoy each of your port days, Denise! Happy travels!