When you leave Memphis, you can go US 64, too. It runs west-east all the way across the state. It's pretty much 4-lane the whole way, too. You can hit the Shiloh Civil War Battlefield if you're interested. I've never camped there, but the David Crockett Park in Lawrenceburg is just off US64. It used to be right on it, but the new bypass misses the whole town now.
If you want to do the Nashville-Grand Ole Opry thing, you can hit I65 north from 64 and easily be in town in about 30-45 minutes. US64 continues east and you can visit Lynchburg - home of Jack Daniels with a great tour of the facilities.
US64 continues on to Chattanooga -- more Civil War battlefields, a great aquarium, and Ruby Falls, among other things, and then on to the Smokies.
You'll have a great time!