Locally, most of the foliage starts greening up around April. Most wildflowers will have bloomed by May. Mind you, though, the higher elevations take a little longer to change.
US 441 that runs through the Smokies from Gatlinburg to Cherokee is a nice drive. There's a good overlook near the top at the TN/NC state line.
The other must-drive road is the
Cherohala Skyway that runs from Tellico Plains, TN to Robbinsville, NC. Nothing but you, the road, and lots of nature. The exact road number is NC 143 whilst in NC and TN 165 in The Volunteer State.
Another fun drive is US 129 along Deals Gap at the TN/NC border. It's known as The Tail of The Dragon; they say because it resembles a dragon's tail. And, that it has 318 curves in its' 11 mile length. Fun drive might not describe it properly, though... helluva drive is probably more correct.
There's a good number of things to see. The Great Smoky Mountains Railway, Cataloochee Valley as French G. mentioned, the Cherokee Indian Reservation, Joyce Kilmer Memorial Forest with its' 400 year old stands of trees, Fontana Lake & Dam, and so on.