We have done something close to the trip you are thinking about several times. This past summer, we headed north out of Florida to Nashville, cut NW to St Louis, then to Omaha and into South Dakota. Headed west to Wind Cave NP, to Mont Rushmore staying at the KOA, then to Devils Tower Wyoming, north to Missoula to visit family for week, then south to West Yellowstone for a week. We stayed at the KOA in WY but it is 6 miles west of town. In town the Grizzly RV looked very nice. At weeks end, we drove through Yellowstone NP and took the loop to the north and east, coming out of the park into Teton NP. Spent one night at Colter Bay in Teton. Then to Vernal Utah for 3 nights, then to Grand Junction Colorado and south to Montrose to stay a week. From there it was to Ouray Colorado for a month, then to Colorado Springs for a week at the Garden of the Gods campground. Then south to Raton NM, SE to Amarillo Texas, to Wichita Falls, on to Fort Worth and picked up Interstate 20 to Shreveport and south to Interstate 10 and back to Florida.
On our 2004 summer vacation, we went north out of Florida to Michigan, toured the UP, took Hwy 2 across into Montana, stopping in the Bad Lands, dropped down to Yellowstone, Medora ND to see the NP there, (T. Roosevelt NP), etc.
A month should make for a very nice trip out west for you. We have also taken Interstate 90 west from the Chicago area, not too far from you. Quick way to get to Wyoming, as we try to get to Alaska, every couple of years, so we are always looking for some "new" and interesting route, to get to the Canadian Border from south Florida.