The beaten path here is remarkably narrow, given all the things there are to shoot. For nature photography, there is Great Falls, the Blue Ridge mountains, the Shenandoah valley, Harpers Ferry, the National Arboretum and others, Rock Creek Park, the falls of the Rappahannock and the James, Roosevelt island, and lots of rolling countryside and little country lanes. Then there are little scenes in and around the Chesapeake and her tributaries. As for more urban locations, Capitol Hill, Alexandria (Old Town and Del Ray), and many other parts of the District are full of lively street scenes and interesting architecture.
The scenery in the East is often more intimate then the West, so you can often find little parks buried in urban areas with plenty to shoot. In addition to Rock Creek Park, I'm fond of Fort Ward and the Mark Winkler Botanical Garden, both in Alexandria, plus a tiny little park down the street. You'll find others just by driving around. You might also want to introduce yourself to the guys at Dominion Camera and ask them.
Welcome to Virginia.