The Frist art museum is really nice, and right now they are having a Chihuly exhibit: http://delevantecreative.com/chihulyinnashville.com/
Cheekwood (botanical gardens) are also great, but go in the morning or evening to avoid the heat.
The Adventure Science Center is also fun, even without the kids, and they have a planetarium show.
Bluebird Cafe, as mentioned, is also cool, but be sure to buy tickets in advance. There is a minimum food/drink purchase ($7 I think) but I would recommend not actually eating a meal there, as the food is not so great.
As far as restaurants, what kind of budget are you on? There are lots of small, cheap places, like the Loveless Cafe as mentioned. Also Open Table, Zacky's, Provence, Calypso Cafe. There is a lot of ethnic cuisine, if you like that. Indian - I like the one on 21st St, Sitar. Ethiopian - Abay, near the zoo. Some good Persian and Thai places too. If you're looking for fine dining, try: The Acorn, Tayst, Flyte, Midtown Cafe, Radius 10, Sperry's. It's usually pretty easy to get reservations.
Have fun!