If staying on the TN side and just getting started with learning the smokies I would recommend researching two areas. Elkmont & Tremont water sheds.
Both these locations allow you to get off the main roads but very close and easy to locate from the Gatlinburg area. Both hold great fishing opportunities, allow for plenty of room to cast your fly and there is enough water that you can fish all week and move upstream and away from other fisherman until your heart is content.
easy to locate, lots of water to fish, plenty of room, and plenty of fish.