Get there by 10AM on weekends if you want a good shot of getting a table and parking. Also, check out the website to read other advice and warnings before you make the trip. I had spotty cell phone reception with an IPhone 4GS. However, my Nike running App, I used it to track distances, worked brilliantly.
Begin at the Visitor's Center and get a map. They are larger and easier to read then the 8.5x11 versions on the web. The people in the Visitor's Center are very friendly so ask them if you have any reservations about the trail.
Before starting, I would take a walk over to the canal and check out the Overlooks so that you get a chance to see the Falls. You don't want to visit the park and not see the Falls. The Overlooks are just South of the visitor center and are marked on the map.
Make your way back towards the picnic area and look for signs for the Old Carriage Trail. If worse comes to worse, make your way towards the restrooms south of the picnic area (they are labeled on the map). There are signs near the restrooms. After the restrooms the path forks, make sure you go towards the right and follow the signs for the Old Carriage Trail. After about 0.1miles, on your right, is the entrance to the Swamp Trail. There is a wooden fence, labeled Swamp Trail, but it is easily missed, especially if plants have grown up around it. So be on the look out.
The trail is pleasant with mild elevation changes and lots of tree cover. After 0.4 miles, you will come to a set of signs. Take a left (the Swamp Trail/Old Carriage Road). If you go right, you will head towards the Ridge Trail on the aptly named Swamp-Ridge Connector Trail. It is has a pretty steep hill to start. You will notice it if you make a wrong turn.
After making the left, continue towards the Old Carriage Road. The Old Carriage Road is wider, as you would expect. After about 0.1 miles, make a left on the Old Carriage Road and take it back to the Visitor Center (about 0.6 miles).
I would not recommend running the trail, because of the rocks and roots.
A nice short trail that is a little better for dogs than the River Trail.