There are tons of both.
Most of the rafting is going to be a little bit of a drive if you stay in Gatlinburg or Pigeon Forge area. But not a big deal--maybe an hour drive out. Totally worth it, though. We love white water. Google white water rafting in Gatlingburg/Pigeon Forge TN and you will find lots of outfitters with information about age requirements.
Most of them have coupons in the flyers in all the display racks in shopping areas/restaurant lobbies. There are also ziplines in the area.
Plenty of free hiking, too. Check in at a ranger station/visitor center for maps to show you which ones suit you. They will list the time/distance, difficulty, whether it is paved or not (some of the shorter ones are), and if there is a visible waterfall feature.
The Cades Cove area is very popular. Also, if you like amusement parks, Dollywood is really a nice one. It isn't all out in the blazing sun (designed with lots of shade) and there are water attractions and a water park. If you go after 4 pm (?), you get free admission the next day as well. But you have to pay to park both times. :(
What part of the Smokies are you planning on spending most of your time? A week could easily be spent in the Gatlinburg/Pigeon Forge area alone. The Wonderworks attraction is loads of fun for the kids, too. (but be prepared, it is loud inside).
---ETA: The farm mentioned by another poster IS really neat. Even has an outdoor "freezer" with explanations of how it worked. :)