We have gone that route before... Nebraska has a lot of nothing... for example, we found the Nebraska National Forest on a map and thought, hey they will have forest service campgrounds there...for sure! I think the forest might have 12 trees! Wow! Anyway, if you have the time, I would travel south from Salt Lake and head to Capitol Reef National Park and then go East from there, travel to Arches National Park, then through Colorado. Lots to see in Western Colorado so explore the map and see what you like! Then from Colorado Springs, Denver, or Boulder head east.... on your way through Nebraska, we have camped at Pioneer Village (worth the stop to see the museum) in Minden, Nebraska (just south of Kearney)... but depending on what time of year, it can be hot... or you will need to watch out for hail storms and/or tornado warnings.
Have fun and post some photos!