I'm a resident so here's my cut-and-paste answer: :)
The Country Music Hall of Fame is fun if you are a country music fan. If you are an art lover, make sure to visit the Frist Center downtown. It's a smaller art museum, but the exhibits are usually outstanding and the building is a beautiful example of art deco architecture. It's really lovely. The Tennessee State Museum is also downtown, and it is awesome and free. There are also lots of bars and restaurants for music like the Wildhorse Saloon and Tootsie's downtown. That is honestly not my scene so I don't know much about it. The Ryman Auditorium (the original home of the Grand Ole Opry) is also downtown. They sometimes have shows there, but you can always do a tour.
For events going on while you are here, you can check
www.nowplayingnashville.com. It shows any type of performance from small to large in the Nashville area.
Near the Gaylord Opryland Hotel, there is the current Grand Ole Opry, the General Jackson Showboat, and Opry Mills mall (outlet mall with lots of restaurants like the Aquarium and Rainforest Café. They also have live music here often). There are definitely lots of touristy things in this area like mini-golf.
There is a lot of history to see if you are interested in that. I love visiting The Hermitage which is the home of President An drew Jackson. It is a really cool place and extremely well preserved (my neighbor is a docent there so I get a kick out of seeing her come home in her historical garb). It is also not far from the airport area. Stone's River Battlefield is in the Murfreesboro which is about an hour away as is Franklin, which also has several large battlefields. You can also visit Belle Meade Plantation or Traveler's Rest Plantation. Cheekwood is a historic home with a lovely botanical garden.
On the wackier side, you can also do things like a bus tour like Nash Trash. I haven't done it, but several friends have and say it's hilarious (and not G-rated). You can also visit The Parthenon at Centennial Park because yes, Nashville has a full scale replica of The Parthenon and the Athena statue form Ancient Greece...lol! There is also a nice, small museum attached to this.
As for restaurants, we love Jack's BBQ and Demos' in the downtown area. If you like southern cooking, Monell's is awesome. It's served family style and you are seated with strangers so be warned in case that's not your thing. The Pancake Pantry, the Bluebird Café, The Loveless Café are "famous" Nashville restaurants, but I'll be honest and say I've never tried any of them. Nashville is also famous for HOT chicken which is super spicy fried chicken. It's also not my thing, but lots of people love it (or maybe they just want an excuse to cry through lunch...lol!).