I live near Disney and I've made the trip to the Smokey Mtns (near Pigeon Forge) with my RV for the past 5 years -- every year! I know about every pitfall along the way. I'll give the best route AND the quickest. IMO
The first trip I used I-75....bad choice because of Atlanta, major contruction all over, and it's a rough ride - bumby. You will saves lots of time taking the eastern board down I-95. (It maybe a few miles longer, but you'll save much time because it's QUICKER. Go to Asheville and take I-26 south to I-95.
Jacksonville is your "problem area" but you can avoid it by taking the I-295 by-pass loop around the city. Go WEST on the loop because you want to go to SR 17. At SR 17 go south again toward Orange Park and Palatka. In Palatka get on SR 19 heading south to Eustis.
(From Jacksonville to Eustis it's a single lane highway going thru the Ocala Forest....your average speed will be 65 mph because it's light traffic....and it's scenic!)
In Eustis you will hit hwy 441 junction....go south on that until you get to Apopka (suburb of Orlando) In Apopka go south on toll road 429 until you come to the Florida Turnpike and go south on turnpike for about 15 miles and you'll be in Kissimmee.
Taking this route will save you 3 to 4 hours (or more) on your trip. If you take this route and can make it to Savannah, Ga, there's a huge 8th Air Force museuem there and many other sites in that city. (8th AF are the guys who bombed the crap out of Germeny to liberate Europe.)
Enjoy your trip!
Phil & Patricia