Braddock Rd. anon here:
Not much in this area. I mean there are some short hikes around Roosevelt Island, and down towards Mt. Vernon or Mason Neck. It's not a long drive to go up to Sugarloaf in MD, or to walk the C&O Canal, though I'd advise starting no closer to D.C. than Swain's lock and walking North. The canal is less of a real hike or something like that, if you want real hiking you'll need to go out 66 to Front Royal and head up Skyline into Shenandoah national park. It's 30 bucks for a vehicle day pass, 50 for a season pass.
You don't want to go now though, the leaf retards are all out there shitting up every trailhead parking lot and scenic overlook. Chances are you wouldn't be able to find a place to park and hike unless you were doing something like the Stony Man loop near the Skyland parking lot, but that one's just under 4 miles and can't fill up half a day, you'll have to move on to something else for it to be worth your while and it's the wrong time of year for that (unless you can go out there early in the day on a week day).
Jefferson and Washington forest is a nice place to hike and camp (actual camping where you hike in with a pack and tent, not this bullshit glamping notion idiots have). I'd say my favorite place and go to though is crossing into WV and hiking/camping over a weekend in Monogahela. Research it a bit, it's excellent. You won't find a ton staying in NoVA, you'll have to go west and escape the limp wristed faggot leftists and non-white imbeciles.