For rafting, I wouldn't consider anywhere else, except for the Nantahala River! Nothing else even compares. Asheville is wonderful (I live in the Great Smoky Mtns, and usually drive the hour to Asheville every weekend...to SHOP

). However, for a vacation like you are looking for, I would stay more towards the Cherokee or Bryson City area. Ziplines are available in the Nantahala Gorge also.
I would recommend whitewater rafting in the Nantahala Gorge, hiking in the Great Smoky Mtn. National Park (in Bryson City or towards Gatlinburg, TN), a day trip to the Biltmore Estate in Asheville, a little fun at Harrah's Cherokee Casino (visit Paula Deen's!), tubing & hiking at Deep Creek (in the GSMNP in Bryson City), gem mining in Franklin (Sheffield Mine), and a day trip through another section of the GSMNP from Cherokee to Gatlinburg (about an hour drive). Cades Cove is a great place to visit with old homesteads, and you can bike around the loop before the road is opened up to vehicles in the morning.
Have fun planning! There is a TON to do here!!
Tiffany