If you ride the Mt Vernon trail, then I would take it south from Old Town (not north to DC unless you plan on taking your time and going in just to tool around the monuments and such). There's often less multi-use traffic on the southern end, but on the weekends, it's not going to be great.
There's some great riding in and around DC. What area of the city will you be in? What kind of mileage do you want to do? Do you want to see the sights or ride to get in a good workout?