About Grand Cayman...
My husband and I were there last week as one of our cruise stops and we rented a Jeep from Cico Avis. We LOVED it! We met an Avis representative at a shop near the pier, he drove us to the airport (5 minutes) and within 10 minutes, we were out the door and on our own...with a map from Avis, of course! We stopped at the Turtle Farm, some beaches along the way, and just toured back roads in our Jeep. I'd highly recommend it! Plus, we had fun learning to navigate from the "wrong side" of the road!
BTW, Avis has a deal for ship passengers. I think the Jeep was $79 for the day and everything was included...full insurance, unlimited mileage, Cayman license for one person ($7.50 for each additional person...worth it so we didn't fight over who got to drive!), and no need to fill the tank.
We were in Grand Cayman last year and hired a taxi to take us around. I think that cost about $50-$60 including tip. Within walking distance of the port, there are mostly shops and we really aren't shoppers so we try to get away from that.
Another thought...we found a more secluded spot on Seven Mile Beach. It is a great beach, but there are usually several ships in port which makes the beach too crowded. We found an odd spot on the beach to relax. We found a cemetary (no lie!) on the beach, parked in front of it, and walked through to the beach. Apparently nobody wants to relax too near a cemetary because it was really quiet there! And, if you want to snorkel there while your buddy doesn't, it'd be a good spot. Bizarre tip, but true!
Have a great trip!!