Jimbo I'm 1/2 a day west of DC but frequently visit our daughter and family in Bristow VA which is just outside DC to the west. There are a lot of things to see and do in that area. You could spend a day or more just in the Smithsonian Museum which is in downtown DC. If you have interest in firearms the NRA building has a nice museum and is there in DC. As mentioned there is a lot of Civil war history in northern VA and some of the battlefields are now parks,
http://www.virginia.org/CWNationalbattlefields/ . Keep in mind that whole area is almost like one big city with plenty of traffic and congestion which is not particularly enjoyable for riding a motorcycle. It will likely be pretty warm there in June as well, if I were visiting the places mentioned I'd probably just rent a car. If I were on a motorcycle I'd head S, W, or N out of that immediate area.
North into Pennsylvania is Gettysburg, a turning point Civil war battlefield, and there's always the Harley factory in York. Central PA consists of rolling terrain and pleasant riding.
South into Virginia is nice, bearing SW into the Appalachian's gets you a little higher and cooler plus the roads are more enjoyable. On a motorcycle I would probably ride SW out of DC toward Charlottesville (University of Virginia) and visit Monticello home of Thomas Jefferson
http://www.monticello.org/site/house-gardens. 30 miles or so west is the beginning of the Blue Ridge Parkway, a 500 mile long road which runs along the top of the Blue Ridge mountain range and is a really nice road to ride a motorcycle on. The entire Parkway is actually a National park, it's free and there's no commercial traffic. There are plenty of cross roads to get off anytime you want. Here's a pic somewhere along it.
West of DC is Interstate 81, once you get a few miles away from it either direction there's nice riding for the most part. Roads heading east/west generally cross the ridges so there are some nice ones zig zagging back and forth between VA and WV.
