Hi ligal, welcome to Cruise-Chat. Jamaica is a beautiful island, but very poor. Please make sure if you're going to take a cab that the driver has credentials. Stay in a group, stay in the tourist areas. They are very aggressive and will try and sell you everything. Learn to say "no thank you" and keep walking. It is one island I highly recommend staying with a ship's excursion, but if you have to go on your own, just be very aware of your surroundings. Don't wear a bunch of jewelry, don't take a wad of cash, and use good street smarts.
Grand Cayman is a very safe island. Seven mile beach is my favorite, and there have been some good reviews of Rum Point. I haven't been to Rum Point, but have been to 7 mile a few times. It is a beautiful beach. There are cabs available where you tender to. Just tell them to take you to a good area on 7 mile, tell them what you'd like to be near and they'll find a perfect location for you. Stingray City is another fantastic excursion there.
Hope you have a wonderful cruise!