Several here have suggested staying in resort style campgrounds so the kids can be entertained. Let me put forward an opposing view.
Most kids get plenty of "artificial" stimulation in their lives already. Taking a camping trip is a good way to let them connect with nature on their terms. Look around at NFS or NPS type CG's and see what the young kids are doing. Often, they on the ground examining an ant hill, poking at a centipede, or building minature cities with the sticks and pine cones laying around.
Kids are focussed much more on things up close. It seems like for most kids, they are 6 or 7 before they really spend any time looking at scenery moving by their window outside the car. So when you plan your days, make sure you spend a fair amount of time at your campsite. Sure, you should take a hike or drive and see some of the scenery that you want to see. But at RMNP, make sure you stop where there is some snow to play in. There are also places with lots of squirrels and chipmonks, but just make sure to keep them at a distance from the kids.
As for your trip, I'd skip hitting Utah altogether, as it will be hot and a lot more time in the car. As Busskipper noted, you might plan a stop in the upper Arkansas River Valley around Leadville/Buena Vista/Salida. Lots of great NFS CG's around there. Spend an afternoon throwing rocks in a lake (under close supervision, of course). Head up to the ghost town of St. Elmo and walk on the wooden sidewalkds. Check the schedule and see if there's a concert in the park downtown Salida.
Finding the right mix of activities for you and your kids is key to a successful trip. And allowing them to play and interact with nature at their level will (hopefully) instill in them a life long love of the outdoors.