I will give you a brief descriptions of the options for traveling from the Knoxville airport to Gatlinburg.
From the Knoxville airport, take US-129 south to TN-35, then turn left onto TN-35 and travel to US-411 or to US-321.
US-411 is the route to Sevierville, where you will turn right onto US-441, which will take you to Gatlinburg. (about 45 miles)
US-321 is the route to Pigeon Forge, where you will turn right onto US-441, which will take you to Gatlinburg. (about 45 miles)
One last option is that when you take US-321, shortly after you pass through Townsend you can turn onto TN-73, which will take you to Gatlinburg through the park. (about 45 miles)
These first three routes are mostly 2-lane roads until you get to Sevierville or Pigeon Forge where you pick up US-441, which is 4 or more lanes all the way from Sevierville to Gatlinburg--the TN-73 route is 2-lane all the way into Gatlinburg.
If you want only 4-lane roads, from the airport you will have to take US-129 north to I-140, then take I-40 west to I-40/75, then take I-40/75 northeast to I-75/640, then take I-640 east to I-40. (I-40 through Knoxville is closed for a major reconstruction.) Then follow I-40 east to TN-66, which will take you to US-441, which will take you to Gatlinburg. (about 75 miles)
You need to be aware that traffic can be very heavy on TN-66 and US-441, especially late Friday and on Sunday afternoons.