If you're going to be in Cherokee, there are a bunch of places you can fish. You can fish the Enterprise Waters in the reservation; for that, you'll need to buy a permit in Cherokee - a N.C. license won't work on the reservation...these waters are heavily stocked, including some very big fish. If you would prefer the wild fish of the park, you'll need either a Tennessee or N.C. license - I would personally recommend the N.C. license...you can buy it online, and print out the receipt as a temporary (they mail you your permanent one). From Cherokee, you can head directly into the park on U.S. 441 and fish all along the Oconoluftee alongside the road. You can also go into the Smokemont area and hit Bradley Fork. However, the stream I would recommend the most is Straight Fork; it is a bit of a drive to get to, but worth it - you have a realistic chance of getting all three species in one day. To reach Straight Fork, you have to get on Big Cove road in Cherokee, and head through the reservation, past the tribal fish hatchery, and then into the park on a gravel road that is, for the most part, in fair shape. You can fish all along the road - from the park boundary upstream for a few miles is your best shot at a brown. Further upstream, brookies start to show up, and by the time you get to the bridge crossing (about 6 miles in), that's pretty much all you'll catch.
You also aren't very far from Bryson City, and Deep Creek and Noland Creek. I've never fished there, but I know the lower stretches of Deep Creek will be packed with tubers this weekend.