As others have said, don't miss Cades Cove. If you hike, there are some great trails there such as Abrams Falls (already mentioned) and Rich Mountain Loop. Outside of Cades Cove, Cucumber Gap Loop is a nice hike, as is Rainbow Falls, Clingman's Dome, Andrew's Bald and more. There is a lot to see around the Cades Cove visitor center as well as around the Oconaluftee Visitor Center. The Sugarlands Visitor center has a nice nature trail behind it. Alum Cave trail is nice as well. There is more to do than you can fit in a week, so I'd probably concentrate first on Cades Cove then Clingman's Dome as well as Newfound Gap Road.
2013 Camplite 21BHS Trailer, Ram 1500 Tow Vehicle