I'm guessing you won't be renting a car? Although, I kind of recommend it.
The thing about Cayman is it won't have tons of crazy things to do that some people require on their vacations. It's simply a really nice high end relaxing destination with great food/hotels and safety.
A couple restaurants to check out are:
Calypso Grill (Get the sticky toffee pudding for dessert)
Morgan's Seafood Restaurant
Restaurante Papagallo-A friend owns a condo across the street and basically lives at this place.
Chicken Chicken (for lunch, in a strip mall, but a locals lunchtime favorite)
The Ritz Carlton Pina Colada is expensive (like $17 I think?) but delicious with a healthy Bacardi 151 float.
Haven't been there in awhile, but Deckers used to have an all you can eat lobster night on Tuesday or Wednesday THINK? I thought it was a good value and the bar made great drinks.
If you're not interested in Sting Ray City. I'd skip it. I still do it on occasion when I''m showing it to friends, but it's always on a friends private boat outside of tourist hours. It used to be an absolute ####-show when it's full. I do know they're cracking down heavily on the number of boats and how much squid they're allowed to bring per boat.
Avoid downtown at all costs when there are cruise boats in. Honestly, I'd just avoid it in general. It's mostly shops directed to cruise boaters.
Camana Bay is a nice outdoor shopping area with a bunch of restaurants/bars.
I'll think of more, or if you have any specific questions I can answer?
Most of the restaurants will require a taxi, or your own rental car. They drive on the left, so keep in mind if you'll want to have a few drinks at dinner.