Clay,
I would suggest Davis Creek between Reno and Washoe Valley. It's a beautiful location in the trees with some fantastic trails to take the kids. I would suggest this as the most kid friendly location. It even has a small fishing pond.
https://www.washoecounty.us/parks/specialty_facilities/davis_creek_campground.phpAdditionally I would highly suggest Fort Churchill in Silver Springs. It is an old Army post from the 1800's and has a wonderful small museum on the site. There is also a historical home just down the road from it. We have used that site numerous times for our Prepper training events. For the kids, there would be a great deal to see and explore with great trails to hike and a nearby Carson River.
http://parks.nv.gov/parks/fort-churchillhttp://parks.nv.gov/parks/buckland-stationLastly, although a little bit of a drive away is Berlin State Park. It's an old abandoned mining town (ghost town) with the additional benefit as being the archeological site of the ichthyosaur state fossil and available to tour and explore.
http://parks.nv.gov/parks/berlin-ichthyosaurBill Engval, the comedian mentions the Ichthyosaur in one of his comedy routines although he says California and it's actually Nevada where it was really studied and documented by Professor Camp.
https://www.youtube.com/watch?v=yH6sE8RMUuUWe have reserved the group campground there for our Ham Radio Field Day event and many of us in the NNPG group will be there in June 23-25 working with amateur radios and honing some of our field craft skills. You would be most welcome to attend with your family and we can include some training specifically targeted to your kid's ages or interests. They would also be welcomed and eligible to work on the radios as well talking to people from all over the country and hopefully overseas.
As a footnote, I always enjoy Boca Reservoir to camp in and if you go, take a half hour and hike the Boca townsite there, just past the railroad tracks to the right when you come into the area. It's set off a bit and I drove past it for years before I was made aware it is there. The park has a trail that tours it up on a bluff. It's a short walk but worth every step to see it.
https://www.reserveamerica.com/camping/boca-campground/r/campgroundDetails.do?contractCode=NRSO&parkId=74128I hope that helps and gives you a few nearby leads that would make for a nice camp out with your family.