Lyds,
Weather should be fine in September in fact very warm in some places, suggest you'll need shorts. Daily tempretures could reach 70+ in Chicago and mid 50's on a night.
After that many it will get hotter along the route touching maybe 80 in the middle of the country (oklahoma).
One place where in could be quite cool is in Flagstaff on a evening, it will be hot during the day but drops significantly once the sun goes down because of the altitude (light jacket/fleece maybe).
Mick