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I only know about Virginia really but you have Shenandoah National Park, which has tons of trails that are popular, some hard, some easy, and then also George Washington National Forest, that's more remote with backcountry camping and some multi-day options as well.
I've been to the Mt. Rogers area near Abingdon, that's really pretty, and then closer to DC is piedmont, and there's lots of little mountains near the piedmont in the blue ridge but they aren't as LUSH.
West of the Allegheny Front is wetter, basically.