What part of Virginia? The geology is very different east-to-west here.
One enormous impediment to longer trips is that virtually all riverbank here is private land, and you have to rent a campsite or camp on an island (usually also private) below the high-water mark. They can certainly be done, though. There are tongues and pockets of National Forest or similar here and there. Most train systems piggyback the river valleys.
There's good day trips on the Cowpasture, Jackson, James, and a bunch of smaller creeks. We tend to less seasonal flow than northern rivers but you'll drag your keel in August. The 60 miles of the Upper James is a good 4-day trip, as is the Middle, but bring your own drinking water due to ag runoff. With one big hop around a series of dams, you can run the whole James for 220+/- miles into the Chesapeake, if you're okay with a tidal section below Richmond. The Bateau Festival in June/July is a delighful spectacle; people make barges out of green oak and run them from Lynchburg to Richmond.
The New River Gorge in West Virginia is rightly famous, though it's mostly whitewater kayaking (one guy built a bateau and ran it). Driving a few hours (depending on where you land) there are good trips in the Carolinas, but now might not be the time to plan them.