I doubt you can fly non-stop from Minneapolis to the Turks & Caicos. Depending on your airline of choice you'll probably have to take a connecting flight out of Charlotte, Atlanta or Miami.

Hint: Use sites like Orbitz, Kayak and Expedia to bring up available flight itineraries and fares- then go to the airline's website when you are ready to book.

We've been flying to Turks from Milwaukee for years. As RoamsAround says above you probably won't find direct flights from Minn. so I would suggest you choose a flight that changes in Charlotte. This is such an easy airport compared to any of the large ones. It will make your flying much more enjoyable (if there is such a thing?) than the huge hassles of Miami, Atlanta, etc.

I'll add that we loved Anguilla, but it is very hard to get to. One of the reasons we started going to TCI is because it was easy to get there, We've returned again and again because of that and the fact that it is everything we love in an island.

Good luck in your planning.

My husband and I live in Minneapolis (no kids yet) and have gone to Anguilla twice and are returning again in early April 2013. We absolutely love it. Wonderful beaches, amazing restaurants and easy to navigate. Getting to many Caribbean islands from MSP is time-consuming, but we learned a trick for Anguilla: Sun Country Airlines has weekly direct flights in the winter months through early April to St. Martin. Anguilla is right next to St. Martin. So we fly into St. Martin and can then take a short ferry (~20 minutes) to Anguilla. The Sun Country flights are just five-some hours and cheaper than the major airlines flights to the Caribbean.
I've checked on T&C flights before, but it's not nearly as easy to get to as Anguilla from Minneapolis. However, keep in mind that Anguilla is a pretty sleepy island with little to do beyond beaches and restaurants. Snorkeling is OK, but not nearly as spectacular as the snorkeling and scuba on T&C from what I hear.