Head to Deep creek. There are nice gravel trails that follow a creek where visitors enjoy tubing. For a little more adventure, drive past deep creek and hike the “road to no where” trails toward Forney creek area– just walking through the tunnel is awesome–make sure you bring a flashlight!
If you don’t mind a drive, I would recommend the Joyce Kilmer Wilderness area. There is a 2 mile loop that winds around some large deciduous trees and a great place for kids. It is located in Robinsville is about 45 minutes from Bryson City.