Welcome to Colorado!
The roads leading to the parks are well developed highways with no issues!
Entering Chatfield off of CO 121 (Wadsworth Blvd) is more likely the entrance you will use. It is the main entrance to the park. It is a few minute drive from the entrance station to the campground loops, but not too bad. Once at the campground, you need to check in at the Camper Services Building prior to set-up, just follow the signs. Stick to the posted speed limits.
Cheyenne Mountain is south of I25 in the Springs. You go south on S. Nevada Ave (turns into CO 115). Just keep going straight, it's a straight shot all the way to the park on the opposite side of Ft. Carson. Once in the park, the roads can be a little steep, but are new and well maintained.
Hope this helps!