I'd agree with the above assessments. You're probably looking at three days of driving in each direction. You might be able to get a night in at the Sand Dunes going or coming. Realize that getting from Durango to Estes Park is a full days drive. Dinosaur and Steamboat are way off of any route one might reasonable take between the other destinations. So head for the Durango/Silverton/Mesa Verde area and make that your destination. If you wanted to add another place, I'd suggest Salida, where you can raft, fish, mountain bike, 4x4 etc. But I sure wouldn't try to get to RMNP, Dinosaur, Denver, etc.
Another vote for Morefield CG in Mesa Verde. Your kids will get a kick visiting the various ruins. One has a series of ladders to climb to add to the experience.
Do take the train. An experience like no other train ride in the country. Get a copy of Cinders and Smoke by Doris B Osterwald. It gives a mile by mile account of your trip and point things out you might not otherwise notice.
Take another day and drive to Silverton. You'll have more time to explore than the 2 hours you'll have for lunch on the train ride. Plus there's lots to see above Silverton. THe drive up to Eureka and on to Animas Forks, a ghost town where they are restoring some of the old houses. Or rent a jeep and explore the higher and more rugged passes. There's an interesting mine tour.
Do a loop out of Durango up to Telluride and then over to Ouray. Take a few hours at the hot springs pool there. Then return through Silverton back to Durango. It makes for a long day, but you'll be passing through some amazing territory.