If I may suggest you get a National Park Pass before the trip to save on all the entrance fees. You might stay in Cedar City and visit Kolob Canyon just a few miles south of there on I-15 as an "appetizer" for the other parks. And don't miss seeing a play at the Utah Shakespeare Festival - one of the finest in the world. You can make day drives to Zion, Bryce, Capitol Reef, and the others from there. On your way through Bicknell, DO stop for lunch and a slice of pickle pie. Really.
Each park will have different rules about how many visitors they can handle and when. Some do require time slot reservations. Go to each one WELL in advance (like NOW) to see if you need to book a time slot or arrange shuttle bus seats.
