Grand Cayman is very touristy but these activities are concentrated around the 7-mile beach, Georgetown, and Camana Bay areas and it is not a college-party destination.
There are other nice locations to stay that are away from the tourist areas but you will need a car to get around (Morritt's on the East End for example, but it is a long drive.)
Grand Cayman is expensive but you can often find reasonable airfares. We like to stay in an apartment or condo so that we can cook some simple meals and then go out to dinner. There are many excellent restaurants and other activities available and Grand Cayman offers great snorkeling and diving. There are also some very nice shore dives which are less expensive.
You might take a look at Sunset House, it offers easy access to both the touristy and more remote parts of the island.
We usually like to stay at the Turtle Nest Inn and Condos in Bodden Town, a small, clean, family-owned/operated venue that is about 10-12 miles from the 7 mile beach and Georgetown and it has a small beach and good snorkeling right out front. Below are some links describing things to do on GC, take a look at them and see if the island might be of interest.
Since you like things like hiking, biking, swimming, snorkeling, climbing, caving, nature, and good food then the recently renovated and renamed Cayman Brac Beach Resort might also be of interest and the diving is even better than GC IMO. The resort is all-inclusive and the food is buffet but excellent. It's a quiet, tiny island and the CBBR is located across from a wetlands nature preserve.
The island also has a parrot preserve and a red and brown booby preserve. For a non-diver there is not a lot to do besides relax and enjoy nature. There are not many restaurants and it is certainly not a touristy or college-party place.
There are also some links about Cayman Brac below. You will need to take an interisland flight on Cayman Air to get to CB from GC, but they do offer daily jet service to and from the island, or you can choose to fly in a (large) propeller plane.
The diving is best off Little Cayman but there is very little for a non-diver to do on that tiny island. Little Cayman Beach Resort is served by Reef Divers who are also the operators for the CBBR and the Cobalt Coast resorts and they provide full valet services as part of your dive package. You would have to fly in a twin otter propeller plane to get to and from LC from GC. There are many fine dive operators in the Caymans.
The Caymans are a clean and friendly destination, and you can eat the food and drink the water without concern. They have a low crime rate, good roads, quality healthcare, and good mosquito-abatement programs. They have a very low incidence of mosquito-borne diseases like dengue, chikungunya, and no problem with malaria.
You might want to check them out and see if they would be a good fit for your family trip, good luck!
Things to do on Grand Cayman (both touristy and non-tourist attractions):
Grand Cayman: Driving Tour - A Guide - TripAdvisor
http://danilew.biz/blog/2014/03/03/grand-cayman-paradise-found/
http://www.scubaboard.com/forums/cayman-islands/437891-topside.html#post6532430
TESTUDO'S SNORKEL GUIDE Cayman Off the Beaten Path
Grand Cayman Restaurants Guide for Cayman Islands Restaurants
http://abcnews.go.com/Lifestyle/skip-cayman-islands/story?id=32807539
http://www.compasscayman.com/whatshot/
http://www.barefootman.com/whahappenin.html
Cayman Brac is a quiet little island; it's a lovely place to relax and enjoy the sun, sand, sea, and nature - but there are not a lot of topside attractions:
http://caribjournal.com/2015/09/15/journey-to-cayman-brac/
Cayman Brac free tours:
http://www.islands.com/free-tour-you-cant-miss-cayman-brac
FYI, a post that I wrote about the difference (for a diver) between the 3 Cayman islands:
http://www.tripadvisor.com/ShowTopi...ween_the_3_Cayman_islands-Cayman_Islands.html