Forgot to say that you will find locals at most places and there should not be many tourists on island at the time you are going. Places to avoid would be the Divi Resort (kind of run down and not good service when we were there for lunch 2 different times). The west end may be kind of hot due to the breeze not being as strong typically this time of year. There is world class diving on STX if that interests you. Also snorkeling the cruise ship pier is supposed to be nice but have not done it myself. Point Udall is pretty (and usually nicely breezy) and there are some hiking trails that go off from there but bring water and do not leave valuables in the car as this is a remote area. Should probably not leave valuables in the car anywhere including where most of us live!
Another great place to eat (also on the East End) is Blue Water Terrace. Pick up a welcome magazine upon your arrival at the ariport. Also Curzan Rum if you like it. Can tour the Rum factory. Interesting and often can meet one of the family owners.