The Butler/Johnson County area is surrounded by the Cherokee National Forest. Within the forest are sections of the Appalachian Trail.
One section which is often hiked in one day, is the Iron Mountain Trail going from crossing Watauga Dam to coming out at Cross Mountain (route 91) about 18 miles later. There are 2 huts on the trail, but by spending the night in one of the lodging facilities listed in the AT (and other) handbooks, slack packing is a great way to cover some ground and still have a comfortable bed and bath! Pick up lodging on the route 321 side, then hike the trail and have another day's comfortable lodging when you come out at Cross Mountain.
From Cross Mountain you continue across Osborne's field over Holston Mountain and then into Damascus, VA.
Damascus is known as "Trail Town USA" for its numerous hiking/biking/horseback riding trails. Again, if you are a fast hiker, this can be done in 8 hours.
And nestled between the mountains is Watauga Lake, great for swimming, boating, and boating.