Billy Goat section A is between Old Angler's Inn and Great Falls Visitor center. You can park at either. On a gorgeous weekend, the parking lot at Old Angler's often fills up. There is much more parking at the Visitor center. It is only $5.00 per car. You also have the advantage here of a museum showing how the canal was built & operated, an original tow boat (currently under repair) and access to a bridge/boardwalk trail that takes you out actually over some of the falls, then to a dramatic overlook.
From there, you just walk along the towpath to the entrance for the trail. There are lots of posted maps. No dogs are allowed on section A.
Section A is the most scenic & dramatic. There are some long stretches over big rocks/boulders so you need to be nimble, but anyone in reasonable shape can do it in regular sneakers. It is challenging, but a stalwart 6 year old can do this trail. There is one steep edgy bit (we've always called it the "Crack of Doom") that is easier to climb going up (from the Visitor Center start point) than down (from Angler's start point.) There are also a lot of places where kids especially like to go "off piste" and scramble over boulders and up cracks. I hiked this section yesterday with a 12 & 15 year old. From the visitor center entrance, along the whole trial with plenty of off trail scrambling, a stop for snacks then walking back on the towpath including the walk out to the falls and a visit to the museum was about 3 hours. With a 45-60 min. drive to and from DC the whole excursion is about 5 hours and left everyone pleasantly worn-out. I would definitely not do the zoo on the same day.
The three sections are not directly connected, and you could go from one to another along the towpath, but I wouldn't recommend it. It would be a long day. Old Angler's Inn is a fairly pricey sit-down restaurant. There is a good assortment of small casual restaurants & shops on the corner of MaCarthur Blvd. and Seven Locks road. Also the intersections of Falls Rd. & River Rd. Or just bring your own food.
Section B is reached from the parking area near Old Angler's inn. Turn left on the towpath for Section B instead of right for Section A. This is a more level trail, mostly along a ridge above the river, with a few rocky places and one very nice rocky promontory. It is mostly through the woods and the trees obscure most of the views when they are in leaf, so it is best in late Fall, Winter & Spring. It is thick with Bluebells in the Spring, and there are a couple of patches of stinging nettles in summer. Total round trip time 1-2 hours. Leashed dogs are allowed.
Section C is reached from Carderock. There are 3 or 4 parking lots. If you turn left when you enter and use the lot near the big picnic shelter, you will see the trail head at the end of the parking lot. Go left (east) and this section is about a mile, level and wide. It is wooded, but closer to the river so there are still good views. There is 1 small rocky bit and one small rocky stream crossing. Go right (west) and the trail is more woodsy, but views are still pretty good. The trail climbs a little then goes along a bluff with some nice views into a small gorge. At the western end there is a popular rock-climbing area. If you go down to the bottom you can watch. It is technical climbing (ropes etc.) but there is also one place where even a 6 year can climb all the way back up to the trail. It looks terrifying but it is easy and safe. You can also park near this area by turning right when you enter and going to the furthest parking lot.
Sections B & C are more places locals go for a casual little hike, I wouldn't suggest someone do all three in one day. Section A is clearly the best, and makes a nice full day.
There are also a lot of really nice hiking trails right in DC in Rock Creek National Park. There is a 6 mile stretch of Beech drive in Rock Creek park that is closed to traffic on weekends, so you walk/cycle/skate right on the road. http://www.nps.gov/rocr/index.htm
The Zoo grounds actually open at 6 a.m. The buildings open at 10:00. nationalzoo.si.edu/default.cfm…