Considering the size of your group, and the fact it's your first cruise (how exciting) I'd highly recommend shore excursions! You can definitely just walk of the ship and hop a taxi in either port, take it to a local beach and find your own way back and it will be much less expensive. But the excursions from the ships are vetted and guaranteed by disney, so if weather or some other issue arises you will get your money back. I've been to St. Thomas many times, i recommend walking around downtown a bit if you like shopping, but the beaches there are great (megan's bay, turtle beach or take a ferry to st johns, it's the best!!!) I didn't do an excursion in St thomas my last trip and ended up grabbing their plentiful taxi's to get around. Tortola doesn't have quite the vast network of public caravans, so taxi's are available but not as much.. The most important thing is making sure you're back to your ship on time.
quick thing to think about as well, if you have AT&t, Tmobile or sprint you should get phone service in St Thomas, which might make getting around the island alone a lot easier.
So if you want to just relax and enjoy a beach or shopping, don't do an excursion. If you want to see as much as possible and get a really unique experience, disney does a pretty good job with their excursions desk, and you can book them once you get one board normally pretty easily.
I also recommend snorkel rentals in Castaway Cay, unless you bring your own. it includes a floating vest, flippers and the snorkel and mask. Also biking can be nice if you like being active, and they do a 5k on the island in the morning if you're into that
Here's the link to disney's carribean excursions--
https://disneycruise.disney.go.com/port-adventures/caribbean/