First, we stopped at the lower parking area near Rock City, a collection of interestingly weathered rock formations. We decided we'd hike out to Balancing Rock, then drive up to the summit and try the interpretive trail around it.
Balancing Rock turned out to be rather further away than we thought, and not too terribly impressive:
However, we did get to see a very neat little dell with multi-colored grass, a great big thistle, a tree that looked like a horse, another sort of tree with spiffy curling bark, and giant pinecones of doom:
Because the hike out to the rock was further than we thought, we were pretty tired when we got back to the car (even with a break for tasty picnic lunch out on the trail, and a further break upon reaching the picnic area near the parking to lie down on a bench and recover for a while). Therefore, we decided to skip the summit trail, and instead just checked out the view from the top and poked around the nifty little Civilian Conservation Corps lighthouse / visitor center there. (It was closed, but the outside of the building was reasonably interesting.)
So on the whole a quite satisfying trip. Can't say I recommend the Balancing Rock, but the views are pretty spectacular and the hiking is nice.