Enjoy your trip
I live in Salt Lake City so I have only been to the North Rim. It is incredible. From what I read, I'm not sure I want to make the trip to the South Rim. WAY too many people and too hot. However, I would go to the south rim if I had tickets to ride the train from Williams, AZ to the South Rim.
As far as Utah is concerned, others have given some good advice. Capital Reef Natl Park should also be on your list if possible. It is in the same general area of the state. Just remember, Southern Utah is very hot in the summer.
If you are going to be in SE Utah, try very hard to get over to Durango, CO and ride the Durango & Silverton Senic RR. It is a trip that you will not regret. Mesa Verde Natl Park is very close to Durango.
Guaranteed that this will be a trip that you will always remember.