This is an amazing hike that's also great for backpacking overnight. You'll hike the Appalachian Trail to Buzzard Rock and pass vista after vista along the way.
Starting at the Elk Garden Parking Area hike the Appalachian Trail south (do not cross the road, follow the trail from the parking area). Hike along the trail following the white blazes and at 2.4 miles you'll reach a campsite on your left. This is a great place to camp overnight if you decide to turn this into a backpacking trip. Across from the campsites is a clearing with a bonfire pit. Continue past the campgrounds, across the gravel road, and enjoy the sweeping views to the south. Hike down into the woods and when you emerge from the other side you'll be in a large clearing and see multiple piles of boulders. Keep hiking until you get to the top of the clearing marked with a wooden post with a white blaze. This is Buzzard Rock (at 3.3 miles)! Spend some time enjoying the other outlooks before retracing your steps back to the parking area.
Buzzard Rock faces west so it is a great location to catch a sunset.
Additionally you can cross the road and hike the short but steep trail 0.3 miles to the top of Elk Garden.