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Here's the thing. It's going to be a minimum of four days each way between Palm Beach and the Colorado Rockies. That only leaves you six or seven days in Colorado itself. So this trip is going to be every bit as much about the journey as the destination. In fact, I'm going to suggest that, due to the young ages of the children involved and the extra time needed to set up and take down camp each day, that you take a full five days each way for the drive and use the extra time to make a few stops each day to let the kids see something of the country, but more importantly just get out of the car and get some fresh air and exercise.
I will also recommend that you take a different route west- and east-bound. The kids probably won't notice, but the adults will and it will make the trip a bit more interesting for you. One route (the most direct) would use I-75 to Chattanooga, I-24 to southern Illinois, I-57 to Mount Vernon IL, I-64 to St. Louis, and I-70 to Colorado. The other would take I-75 to Lake City FL, I-10 to Mobile AL, US-98/US-49 through Hattiesburg to Jackson MS, I-20 t Dallas-Fort Worth, US-287 through Amarillo and eastern Colorado to Limon, and I-70 to Denver. You can, of course, use either route in either direction.
Generally speaking, national park camping is 'better' in that you are camping at a major national scenic or historic site, but their camp sites tend to fill up early. Most of the more popular parks are probably already fully booked for this summer. While they do have cancellations and hold some sites open on a first come/first served basis, you really can't count on either of those if you can't get a confirmed reservation before your departure. State parks tend to be more numerous and somewhat cheaper, but again reservations are recommended during the busy summer season.
In Colorado itself, you can build a very nice loop around Rocky Mountain National Park, Leadville, Great Sand Dunes National Park, and maybe Florissant Fossil Beds National Monument, that will make the most of your time there
AZBuck