12 day Carribean cruise - Exploring the island without excursions?

My wife and I will be doing a 12-day Caribbean cruise in December on Celebrity. The stops are in Punta Cana, St. Croix, Phillipsburg (St. Maarten), Roseau (Dominica), St. Georges (Grenada), Castries (St. Lucia), and Basse-Terre (St. Kitts).

This is only our second cruise. The first time we did the Greek islands. When we did the islands, we usually rented a scooter or 4-wheeler and explored the islands on our own. We would prefer to do something similar and not have to use the ship's excursions. We feel they can be overpriced and a bit too slow for our taste. We prefer to explore on our own.

Do most ports have scooters? Is it safe to do this?

Are there any must see things on each of the islands? My wife and I went through the excursions and wrote down some of the things we would like to do (kayaking, snorkeling, cenotes, waterfalls, and lounge on the beach).

Thanks in advance!

Tony