For something a little out of the ordinary visit Hollywood Cemetery. Belle Isle in Richmond is also nice. It is accessible to pedestrian and bicycle traffic via a suspension footbridge that runs under the Robert E. Lee Bridge. I haven't been there in years, but when I was there there was an abandoned power plant and other abandoned buildings all accessible and not fenced off.
I used to walk around Belle Isle a lot. It's very historic. There was a prison camp there during the Civil War. The Tredegar Iron Works is just a few hundred yards away and worth visiting.