I love tooting the horn for my hometown - Alexandria, VA, a very historic city, older than Washington, D.C.
Lots of things to see and do -- and great places to eat, as well. Alexandria is the fun side of the Potomac River.
Mount Vernon, the home of George Washington, is just under ten miles away, straight down the George Washington Parkway (known as Washington Street in Alexandria). Mount Vernon is located in Fairfax County. (In Virginia, you live in a city or in a county.)
One can catch a river cruise to Mount Vernon, but it is easier to drive there. Parking is free. http://www.mountvernon.org/
The Mount Vernon Inn is a nice place to have lunch or dinner. Another lovely house to visit is Gunston Hall, the home of George Mason. The house is not too far from Mount Vernon, as Washington and Mason used to travel by boat to each other's homes. By road, the trip is about 9 miles.
Located on the Woodlawn Estate is the Pope-Leighey house, a Usonian house designed by Frank Lloyd Wright. http://popeleighey1940.org/
Woodlawn, which was once a part of Washington's estate, is now run by the National Trust for Historic Preservation.
Fine dining since 1770 - Gadsby's Tavern. A fine establishment, great food, but a bit pricey. Rather partial to Gadsby's because they catered my wedding!