Sounds like an adventure
Once you're on the Mount Vernon Trail you can follow it all the way though Alexandria to Mount Vernon. Very pretty. You can cross at either the Memorial Bridge or a little bit farther down at the 14-st bridge.
If you go the other way over the Key Bridge, you can catch the Capital Crescent Trail, or even the C&O Canal Towpath (if you have the appropriate tires for the unpaved but well-packed surface.)