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Washington D.C. has a lot of public parkland.
Usually when I am in town my favorite trail for biking (certainly suitable for walking as well though) is the Anacostia Riverwalk Trail as it runs along a placid scenic river and you are surrounded by trees on both sides and even water on both sides in a few stretches. You are bounded by the highway on one side and water on the other and cars aren't allowed and it is a paved trail so it is ideal for biking.
There's also some other trails in the city such as through Rock Creek Park but that tends to be more crowded.
To the south and east there are many trails through gently rolling tidwater hills and along the shorelines of rivers and the Chesapeake bay.
To the north and west there are mountains about an hour's drive away (catoctin/sugarloaf mountain), fairly high (for here) mountains 1/5-2 hours away (2500-3500 ft elevation) in Shenandoah national park, but again that tends to be crowded during summer, fall, and weekends. If you are willing to drive 2.5-5 hours there's AMAZING trails all through West Virginia and Pennsylvania in the Alleghenies and for me those are the best nature in the area as those are the lushest sections of deciduous forest.
