Welcome aboard, Carolina!
I highly encourage you to take a look at Disney's Port Adventures. When you click on Port Adventures, you can narrow it down by age on the side. Plus, prices are low (or sometimes free for ages 0-2)!
However, sometimes it is fun to simply head off the ship on your own to explore. If you're looking to head to a local beach on your own, of those three ports, I'd feel most comfortable recommending Grand Cayman. Grand Cayman requires tendering. It'll anchor out in the bay and smaller boats will transport guests from the ship to the dock and back. If you head off early in the morning, you'll have plenty of time to do a little shopping in town and head to a local beach. There are buses and taxis just outside the secured port area that you can catch to head down to Surfside Beach, a part of the beautiful Seven Mile Beach. Expect to pay around $5 per person each way. There, you can rent chairs and umbrellas for around $5 each too. It's a great spot to relax and play in the sand and sea with your kids, other water activities are available, and food spots are right there. Just remember to watch the time so you're back to the port with enough time to wait in line to catch the tender boat back.
In Cozumel, my family loved our
Cozumel Beach Break adventure at Playa Mia. With not only the beach but a fun water park for kids, even little ones, it's perfect for your family. Now, in Jamaica, I wouldn't recommend heading out to a beach on your own, but I highly recommend taking a Port Adventure that includes a visit to Dunn's River Falls, one of the most famous spots in Jamaica! Perhaps the
Dolphin Cove and Dunn's River Falls adventure!
Have fun planning, Carolina!