The Cayman Islands have their own currency. $1.20 USD = $1 Cayman. USD is widely accepted so there's no need to exchange your money before traveling, but you'll get your change back in Cayman Dollars. It's a good idea to have cash for tips and for cabs. All major credit cards are accepted in hotels, restaurants and local shops. The local currency is the Cayman Islands Dollar (CI$) but transactions using non domestic credit cards will be charged in US$.
The Caymans are a pretty expensive destination, but well worth it. There is a 20% tariff on all goods that you'll notice on reservations and receipts. It's how the local economy is supported, but always a good idea to plan for that.
There are multiple grocery store options on the Island, where you will find pretty much anything you will in the US. We usually walk (about 15 minutes) to the Cost U Less or Foster's. They let you take your grocery cart back to your hotel or condo and collect them around the Island daily.
18!
There are cabs that run along the main stretch of the island really frequently. You can just wave them down, but we usually meet someone we love and get their card and call them if it's a longer ride. They're very safe and friendly as can be.
You're able to rent a car, but we never have. You need a renter's permit for $16 KYD which you get at car rental businesses. You can rent cars starting at $18/day and need to be 21 to rent, 17 to drive. They drive on the left!
You'll need an international plan to make calls without wifi, but most places have free wifi to use. WhatsApp/FTA will be the cheapest way to communicate. Wifi is readily available.