Nashville, Indiana is located very near Brown County State Park. Brown County State Park is probably THE most popular and visited and loved state park in the entire state. They provide all sorts of activities, features, and trails.
Nashville is not too far from French Lick, Indiana. You can take a train ride at the Rail Road museum, visit the French Lick Casino (You don't need to spend money gambling), and remember, this is the childhood home of Larry Byrd.
Bloomington, Indiana is not that far away either. Bloomington, home of Indiana University. It'w worth a driving tour of Bloomington and IU.
And not to mention, Hoosier National Forest is nearby too.
Nashville is full of little shops, and a fun little town to visit. There are a number of private campground in the same area, and if you do a simple Google search on the Internet for "Things to do in Nashville, Indiana", you'll get quite a variety of ideas, pick what interests you, there's a lot there.
Good luck and enjoy your trip!