I live in Virginia, in the Washington, DC, suburbs. Basically, my feeling about sightseeing is that you have to walk it to do it. If you're going sightseeing with me, be prepared for walking.
Virginia has lots of pretty places near me, including the Blue Ridge Mountains, Shenandoah River, and several well-kept Civil War battlefields. All are worth seeing, and I particularly recommend a hike on the Appalachian Trail from Rt. 7 (near Berryville, VA) south to a beautiful rocky overlook. Another gorgeous (and vigorous) hike near me is called the Billy Goat Trail, and it's in Great Falls Park in Maryland.
Then there are the museums. Washington has many world-class museums and most of them are free. There are many Smithsonian museums and I like them all, but my fave is the Air & Space museum, where you can see the Wright Brothers' Flyer and touch a moon rock. The National Gallery of Art is superb, as is the Portrait Gallery, the Postal Museum, the Building Museum and the National Zoo. Of course, the Capitol, a walk past the White House and a visit to many of the monuments are also essential. All the places I mentioned are free.
One final must-see in DC for cheapskates like me is the free concert every day at the Millennium Stage at Kennedy Center. Every evening, there is a free concert or performance, everything from a capella groups to brass bands. I most recently saw a Czech girls choir on the Millennium Stage and it was surprisingly good, despite the fact that I speak not a word of Czech.
Enjoy the trip!
--Bob