Maybe someone else can offer up some advice on Memphis and Nashville. Neither has ever been my favorite, although the Vanderbilt area in Nashville is pretty cool. Memphis mostly sucks and I am not a Grand ol Opry / Gaylord Hotel type of guy so I never hung out much in Nashville.
Once past Nashville you need to spend a day around the Plateau. The Plateau runs from Nashville to Knoxville and from Kentucky down to Chattanooga. The Cookeville/Crossville area is really cool because the geology (Karst) lends itself toward big waterfalls and caves. Al of the Cookeville area falls are great, best are Cummings Falls, Virgin Falls, Burgess Falls, Stinging Fork, etc. Cummings falls has a fantastic swimming hole with real rednecks running everywhere. It is great. Virgin falls is cool because the river appears from underground, drops a hundred foot falls, then disappears back underground. There are 50 to 250 foot waterfalls everywhere, just google them. Hiking in the plateau is wonderful, no bears, few snakes, etc. Just millions of acres of rivers, creeks, falls, gorges etc.
Knoxville is OK, worth a bit of time in the old city or market square. If you are a mountain biker then dirty south XC and haw ridge DH are really good. Pigeon Forge sucks. Gatlinburg is better but still all tourists. Dollywood is surprisingly fun, get there right when the gates open and hit the roller coasters fast before the lines start.
Chattanooga is a better city than Knoxville but doesn't line up with spending time in the smokies. Cool river side downtown area, great mountain biking, Ocoee rafting, etc.
Next up is the smokies. If you are in the spring or early summer you gotta see the fireflies. There are millions of them and once your eyes adjust, there is nothing like them in the world (except one area of China). Here is how you do it without dealing with the touristy BS, head through Gatlinburg past the Sugarlands visitor center. Drive about 8-10 miles up and the road (441) will take a hard left and go over a bridge. Straight ahead is a campground called the "picnic area" which is located on the upper west prong (one of the best kayaking runs in the world). Go over the bridge and look for a roadside parking spot (they lock the picnic area gate at dark). Park on the road and walk into the picnic area and set up shop at one of the picnic tables. About 30 minutes after sunset the fireflies will start up and they will synchronize shortly thereafter. Unreal show. It will be like nothing you have ever seen. Watch out for bears, there are all over that campsite but they will back down if you stand up and yell at them. Seriously don't freak out about the bears, they are everywhere and will come very close to you if you let them.
Next up - Clingman's dome trail. Continue past the picnic area to the very top of the mountain another 15 miles or so up. You will pass the needles trailhead and road prong on the way up (wonderful hike). Go to the clingmans dome parking lot and park. Walk like you are going to the observation tower but there is a trail branching off to the left before the walkway to the observation tower starts. Follow this trail a mile or so to the best overlook in the smokies. Perfect field of spongy grass overlooking western North Carolina and Fontana Lake. Hard not to fall asleep in the sun here.
So once back in Gatlinburg head up to I40 between Knoxville and Asheville. Take Harmon Den exit and head north on the gravel road. About 10-12 miles up it will come to a T intersection where you will go left. Go another 4-5 miles and you will come to Max Patch. Hike to the top of Max Patch (easy but straight up) and enjoy the view. Figure out which mountain is Mt Sterling and which is Leconte. Do the 3-4 miles hike around Max Patch. Hope the fields of blackberries are in. Rather than going back the way you came, continue past Max PAtch down the mountain and when you come to road hang a right for 10 or so miles to Hot Springs NC. Cool little hiker town.
One other cool activity is to bike around cades cove in the smokies, but you have to have bikes. On wednesday and sunday mornings they don't open the road to cars until noon or so and you have the roads to yourself. It is 11 miles around the asphault but gravel roads cut off parts if you get tired. There are some small hills but nothing big
All of these can be done with kids, if you want some longer stuff or more adventurous stuff let me know and I can give you some ideas. I can also hook you up with the best rafting, biking, hiking, restaurants, etc.