10 hours via I-80, I-76 and I-70 to Glenwood Springs CO. Just north of the Interstate is the world's largest hot spring fed swimming pool. It is about 10 hours from GWS via Denver and Gillette to Devils Tower. After leaving Devils Tower you could visit the Badlands and Mitchell SD (Corn Palace).
The loop is about 2000 miles.
This won't work for a two day trip. How many days do you have planned?

Funny you suggest Glenwood Springs - we went there the last two years! And by the way, it is a great family destination. We've got about 8-9 days planned for the trip. We did talk about the Badlands also.

How do you feel about ten hours in the other direction? Northeast Indiana has a really lovely state park just north of Angola. Pokagon State Park has tent camping, a large lake with sandy beach, boat and paddle boat rentals. There are lots of trails and horses are also available.

Plenty of family activities are within a half hour's drive including: mini-golf, go carts, water park, buffalo farm, classic car museums and Shipshewana, an Amish village with shops and restaurants where you can take a buggy ride.

Just a little farther away is St. Joe, where your family can take a free tour of a pickle factory and sample pickles and peppers at the end. And Fort Wayne with its wonderful children's zoo and Science Central.

If you're coming in the summer you might even catch one of our wonderful county fairs!

The Amtrak westbound California Zephyr stops in Omaha about 11PM every night. You could get off in Glenwood Springs after the amazing views of Glenwood Canyon which I consider to be the best daytime scenery on any Amtrak route.
If you could stand 36 hours on the train you could ride all the way to Reno NV and rent a car to carry the camping gear.

Minnesota

You can make it to Duluth in 8.5 hours, from there head up the North Shore of Lake Superior. I think there are 7 or 8 state parks within 3-4 hours of Duluth along the shore.

You could also go to Ely and Bear Head Lake State Park, or even Voyageurs National Park (but that would require a boat to really enjoy the park).

Plenty of other state parks in Minnesota as well, anything from prairie to lakes, river bluffs to northwoods.

You bring back memories of many summers of family camping with kids...tent, fold downs, trailers. We've been to the places you mention out west. General observation: state parks in Missouri are excellent. Ditto Arkansas, Indiana and other states not so far away. Yes, there are things to do and see as well, e.g., caves in MO, hiking in AK. I could get specific but why not search on your own. Wisconsin Dells is very nice with a boat ride and good campsites. Closer is Lake Okoboji in NW Iowa and parks in MN.

Now then, one surprise we had was in NE en route to the Black Hills. Ft. Robinson St. Pk. has good campsites with interesting things to do, swimming, history, jeep rides into the hills, dramas, etc. Nearby is Chadron St. Pk. which had an encampment.

Bill in Boston

What month?

I lived near Devils Tower for 2 summer. It's an awesome park and the hiking is really great. There is a lot to see and do in the area. The Bear Lodge mountains are nearby and are very remote. This small mountain range is never advertised. It's mostly national forest. We picked a lot of black berries in the summer. The black hills are close too.

Theodore Roosevelt NP is an undiscovered gem of a place. I like to describe it as "Badlands NP without the crowds." Although it's not breath-taking, jaw-dropping beautiful like Yellowstone, the isolation, and the wildness of the wildlife, are real. If your family can't amuse itself by spending time in the outdoors, this is NOT the place for you -- nobody will be up there to entertain you if you get bored.
One last thing -- if you do decide to go to TRNP, read up on the man after whom the park is named. If TR hadn't come to this area, the degree of natural preservation in the USA would have been FAR smaller.

I should add that WhereAreWe's suggestion about parks along the shores of Lake Superior is also excellent.