How old are your kids? That might make a difference as to what people might suggest. It sounds like you have a week to spend out in the Rockies, plus time to travel back & forth (total of another week?) PghBob has some good suggestions for you.
I might suggest heading straight to Colorado. I-80 will have more interesting things for kids, some pioneer-themed parks and some state parks. For example, Nebraska has the Stuhr Museum of the Prairie Pioneer, Ft. Kearny State Historic Area (with Kearney State Rec Area nearby, decent CG there that your rig would fit into), and a delightful museum with great dioramas about the pioneer trek in a big arch over I-80 near Kearney. Ft. Kearny is a real fort - though it won't look anything like the stereotype - built in the late 1840s, with some of the buildings still standing or rebuilt, the ruins of others, and lots of artifacts.
From I-80, take I-76 from Ogalalla, Nebraska to the Denver area. Rocky Mt National Park is worth seeing, for sure. Other places in Colorado your family might enjoy: Colorado Springs (AF Academy, Garden of the Gods, etc.), river rafting (if the kids are old enough) near Buena Vista, and if it has recovered from a wildfire, Royal Gorge near Pueblo. Some decent CGs in those areas. Colorado has some nice state parks too - we stayed at Mueller (west of Colo Sprgs) and Golden Gate (Boulder-Golden area). Reservations recommended.