The Nantahala Outdoor Center hosts a variety of things to do in the Bryson City area, and is pretty much considered the hub of activities for the Nantahala Gorge.
guided raft trips
raft and duckie rentals
Appalachian trail goes through here
there is also a Hostel there for thru-hikers or anyone else who wants to visit
flat water paddling
a few restaurants
The Great Smoky Mountain Railroad also stops here for lunch
And if you just want to watch, you can walk up to "Nantahala Falls" and watch rafters fall out of their watercraft. Tons of fun to watch and laugh at people.
Adventurous Fast Rivers is a outfitter for rafting or kayaking on the beautiful Nantahala River. And in my opinion, they're the best bet for whitewater adventure. Their rafting and funyak (inflatable kayak) trips are less expensive than Nantahala Outdoor Center (NOC) and being a smaller, family-owned company, you'll get better, more individualized service. We last visited as a big group of about twenty and rented funyaks for a trip down the river. They dropped us off at the National Forest launch site near the power plant and paddled downstream to the take-out just before the NOC. Several people in our group were experienced in whitewater, so we didn't need a guide, but if you're new to whitewater or would feel better, they also provide guided trips. Along the river are many class II rapids, including Patton's Run right at the beginning, Delebar's, Quarry, Whirlpool, The Ledges, The Bump, and at the end is the class III Nantahala Falls. Nantahala Falls is definitely the most challenging, and almost half of the people in our group fell out on this one. It is possible to do a shorter trip and skip this one if you're not comfortable with class III. All paddlers and rafters stop at the outfitters along the river before finishing up, a little ways before The Bump. This allows them to give instruction on navigating the falls as well as notifying them that you will be finished soon so they can pick you up at the take-out. As a warning, Nantahala River is extremely cold year round. It is a dam-controlled river and all the water comes out of the bottom of Nantahala Lake, so its 45 to 50°F year round. We rented wetsuits from the outfitters - a very wise investment. If you'd like to get an idea of what to expect, I made a GoPro video of paddling the river; you can see it here.