Use a weather website like
"Weather Underground" to look at historical average, high, and low temps for the place and month you contemplate traveling. For example, if you look at Moab UT and scroll down, on the left you'll see (what this month says) "December Calendar View"... click on that. Then use the arrows to change to the month you want to be there, and see what the weather was like last year plus the record highs/lows. If you do this for several cities along your trip route (keeping in mind that elevation changes will play a big role in temps), you can figure out what your weather is likely to be like for your trip. Don't like what you see? Map out a different month, earlier or later.
Short daily drives with plenty of stops for scenery, museums, and other fun things will make the trip pleasant and memorable. Don't be like my dad, who had to drive nonstop from MI to FL and nonstop back, with the idea of maximizing the time spent in FL; that travel style was not fun!
On your way, I'd recommend some leisure time in Arkansas at Petit Jean State Park; the bear cave, the turtle rocks, the big waterfall, the high overlooks, some great hiking trails, and educational evening ranger talks in the amphitheater will make the kids think this IS the destination. Here are my granddaughters at Petit Jean on the Bear Cave trail.
You can see a ton of different places in Utah without feeling like you are doing a ton of driving. Zion to Bryce to Capitol Reef to Goblin Valley SP (your kids will love playing in this state park!) to Arches to Canyonlands. And from there you are not far from Telluride, Ouray and the Million Dollar Highway to Durango; if you have to drive out or back, why not take the scenic route! Weather permitting, of course... snows come early in those mountains.
