If I could rename just one of the trails in the DC metro area, I might pick Section C of the Billy Goat Trail. There's no rock scrambling, so it's the least billy goat-y of the three billy goats. Instead, I think of it as the Paw Paw Trail. Paw paw trees line much of the route, and we've reliably found ripe fruit here every September for several years.
Whatever it's called, this is an easy 1.6 mile trail (plus a mile back on the towpath). It's accessible from Carderock Recreation Area off the Clara Barton Parkway (see the park service map). From the westernmost parking lot, you'll start on a boardwalk and then follow the blue blazes as the trail goes off to the left. There are lots of little side trails in this area that have been bushwhacked by rock climbers. Considering how many great rock faces to climb and boulders to boulder on there are around here, the trail is remarkably un-rocky.
You'll go downhill and cross a couple of streams. By the second stream there's good access to the Potomac, with a wide open view of the river.
Out of the three Billy Goat sections, this walk back on the towpath is the least interesting -- and the closest to the road. You'll hear, and even see, cars on the Clara Barton Parkway for most of the way back. If you're up for the extra half mile, you might do better to just retrace your steps and take the Billy Goat back to where you started.
See also: Section A, Section B