Having been to Grand Cayman many times, I can tell you that this is a very safe port. It's a British commonwealth, so it appears to be very 'Americanized'. The prices for items are higher than other ports, but the quality is better.
You can sign up for shore excursions either through the cruise line or through an independent company, of which there are many. One of them we've had great luck with is
Welcome to ShoreTrips® - Your Guide to Shore Excursions
When you get off the tender, you can walk to either one of two scuba diving shops, one on the right or one on the left. Both are within two blocks of the pier and offer snorkeling right there. So if you just want to do some snorkeling and don't want to pay for an excursion, both of these are a wonderful way to do it and save a great deal of money.
It is easy to get around the island, including going to Hell, which is nothing more than a tourist trap, or to the Turtle Farm. However, taxis are not cheap. There's no need to take an excursion to get there, because public transportation is cheap and plentiful. Ask anyone where the library is, which is about 3 blocks inland from the pier, and you'll find vans waiting to take you anywhere you want to go. Price was $2.50 per person (last time we were there) for one way. Just tell the driver where you want to go. Taxi fare is about 2 to three times what it is for public transportation.
The Turtle Farm is very interesting.
Snorkeling with the stingrays is awesome! Unfortunately, you need to take an excursion to get there. But I would recommend an independent tour for this as it will cost you about the same, but will be a whole lot less crowded. The cruise line excursions are like 'cattle call' - very crowded and not alot of fun with all those people.
If you want to spend some time at a really nice beach, Seven Mile Beach is wonderful - beautiful warm water and white sandy beaches. We have a spot we particularly like, which is called Calico Jack's. It's right by a public transportation stop, so it's easy to get to on your way back from the Turtle Farm.
Shopping is very good in Grand Cayman, but stay away from the Tortuga Rum. It's not a very good rum and the rum cake is equally unappealing. There is some wonderful rum cake available - try them for free till you find one you like.
Let me know if you have any questions or need additional information.
Pete