A couple of things I'd recommend looking into before the trip, get a day hiking book, check out
this GSM hiking website, and get the
National Geographic topographic trail map.
For hikes we enjoyed, there are several excellent ones along the Roaring Fork Motor Trail which is accessed from downtown Gatlinburg at stop light #8. The motor trail is in itself a great experience. Lots of places to stop but parking is at a premium. Best times are early morning.
Definitely spend some time not just driving, but stopping at the sites in Cades Cove. Lots of history there and several good hikes as well.
There is an excellent hiking shop called the Day Hiker in Gatlinburg in one of the "alleys". Prices are reasonable too, also the best T-shirts in the area.
I know you don't want touristy, but the free "moonshine" tasting is a hoot. There are at least 5 of these places in downtown Gatlinburg.