For those who like the outdoors, there’s some good nearby hiking and views at Beaman Park and Harpeth River State Park.
For those who like the city, Nashville is just 20 short minutes away. There are plenty of things to do for everyone in Nashville, such as Broadway (21+), alcohol tastings, plenty of amazing breweries, Music City attractions, Grand Ole Opry and Ryman Auditorium, plenty of museums about history and music (Tennessee State Museum is free admission), Nashville Farmers market, Centennial Park (The Parthenon), Vanderbilt and Belmont University are downtown, as well as the Frist art gallery.
For those who are Catholic (or not) the Cathedral of the Incarnation is downtown and is beautiful. St. Mary of the Seven Sorrows is the oldest Catholic Church in Nashville, and is also really beautiful church to visit.
There are lots of hiking and state parks east of Nashville as you get closer to Cookeville (1.5 hour drive), could be an option as a day trip.
If you go to Greenbrier, the bride and groom highly recommended a wonderful small town restaurant called Greenbrier Pizza and Pub. They serve pizza, sandwiches, local beer on tap, as well as some amazing other food options.