You are about where I was at this time last year having a desire to get into shifters and include my kids. You can do both! There are a at least four options out there in the NoVa area.
One is to get involved with the Woodbridge kart club. They race on full size road courses like Summit Point, although most of their membership is not focused on shifters so much. They do have plenty of kids classes from what I saw although I have not joined this club or tried any of their events. So this is fairly high speed, wheel to wheel karting in generally enduro style (laydown) machines.
If you’re interested more in sprint track style karting, there is a track near Baltimore called Sandy Hook Raceway that runs sprint track races pretty regularly. They don’t have a shifter class currently, but they are pretty amenable to whatever you bring there. If they were a little closer, I’d probably be a regular there. Short track, and super fun to run on. There are other similar tracks, but I’m pretty sure they’re all farther away. VIR has a nice kart track (at least according to their site) which looks like fun, but thats a bit of a haul from Front Royal.
Third, if you’re interested in indoor karting, AllsportsGP (Sterling, VA.. google it) is fun. They have a driving course and league for kids (check their site, there is an age limit I think). You won’t be able to run a shifter there, since its a small indoor track and you rent their karts. It does keep things cheap though since you don’t have to buy alot of gear. Its a nice way to get started. There is also Summit Point Kart, which is the outdoor equivalent to AllsportsGP where you rent their karts and run on a dedicated kart track at Summit Point. Tis a shame, but I don’t think they have any organized “run whatcha brung” racing on that kart track.
Finally there is autocross. Not something that you might think of at first, but autocrossing a kart is a great way to go. The SCCA has at least 7 events from March through October, generally run at Fedex Field. There is a decent contingent of shifters there (typically 4-8) that run regularly, and a similar number of junior karters too. Its a family oriented environment, and the folks are great. They run a kart school for kids at the start of the season. There is also Autocrossers Inc (another local SCCA chapter), who run a similar autox series, but I have not been to any of their events and I think its less frequented by karters in general.
One other thing. You’ll likely get some replies that say something like “a shifter is a handful for someone new to karting, start with a fixed gear kart”. They are correct in that the learning curve on a shifter IS steep, and if you dive straight into a road course in a 100+ mph capable machine, you can get yourself into a heap of trouble. With maturity and a sensible approach though, you can jump right into one in my opinion. Something like autox is pretty kind in this scenario since its not wheel to wheel, and you can take it at your own pace without any drama. Thought I’d mention that since I hate it when the first response to someone interested in shifters is sometimes a major turn-off.
That’s an introduction … feel free to fire away if you have questions.