Wherever you go, and especially if you somehow end up passing through MO, I'd recommend floating a river on a raft at least one day.   Great way to work together as a team and to bond, be even better if they could drink beer with you.  

In all seriousness. Good camping/road trip in "America" (very general/vague) isn't just like all these little countries in Europe or Middle East for that matter.

Every region offers something different, geography and climate change across the US.

Research what you really want to do because you will miss most things on a 7 day camping trip.

It would take you multiple years to experience Americas different landscapes, wilderness, and regions.

We take our camping seriously and take 2 week long trips twice a year. So very many awesome places to see that I hesitate to suggest one but I will give some tips. Any National Forest ( not National Park) has what is called dispersed camping. Basically you can camp off of any Forest road. The caveats are you cant be within a certain distance of established campgrounds, your camp must be a certain distance from a lake or stream and there is a certain distance off the road you can take your vehicle. These distances vary depending on the National Forest, just call or stop into the Ranger station. While your at it pick up whats called an MVUM or Motor Vehicle Use Map. It will show every forest road, logging road, etc and what type vehicles are allowed to travel on them. Shooting is legal on National Forest land. Many State Forests also have dispersed or as many call it primitive camping with more or less restrictive rules. Also, many of them have cabins ranging from primitive to super nice available for rent (book online at ReserveAmerica).Especially on some of our more extended trips where were moving around alot we rent a cabin for a day or 2 just to take a breather and relax.

Thanks for all the replies.

Some great places already posted here that I have never heard of.  The perfect spot would be off of a logging road where I can take my truck and the gear with the nephews. Set up base camp, tents and a fire. Will bring .22's with cans for them and fishing poles.

Sounds like the Boundary Waters is beyond his skillset.

Southern Ohio has some good public land that might work for car camping, but camping near water may be tough to get solitude.

At one time weren't you renting a cabin up there?  IRRC you bought a place?

As far as eastern mountains go, the Mon was already mentioned.  Nearby is also the GW NF and Jefferson NF.  Near me in the Jefferson, there are umpteen forest trails and lots of campsites.  Even a nice shooting range or two.

There are thousands of options.

What geographic area are you starting out in?

They kids live in the east coast and midwest but I will gather the information and let them decide on the location. I am leaning montana or wy. But will leave it up to them.

Superior National forest is another. Guns are definitely good to to there. Miles upon miles of logging roads with lakes everywhere.

Not WA. You need a handful of passes to do anything around here. Just to pull into some of the rest areas you need a pass. You need one for state land. Another for Federal. Another for parks. Boating pass. ATV pass.  It's like the Russian mob runs this state and needs their hand greased for each recreational activity. One reason is WA has screwed their hunting and fishing participants so hard and long that less people are doing it. That adversely affects their usual cash cow and their budget.

Like old sports cars? Check out the HSR and SVRA racing schedules. Nothing is more fun than camping under the trees along the fence for 4 days watching a bunch of filthy rich old White Men flogging the piss out of priceless museum pieces.