In reply to Rog Saxton:
As others have said, the normal place to fly into is Salt Lake City. There is excellent climbing in the canyons just west of the city--granite (Little Cottonwood), quartzite (Big Cottonwood), and limestone (American Fork, some high up in Little Cottonwood)--depending on the canyon. September can still be warm, but there are always shady options in the canyons. Routes are primarily single pitch but there are longer ones as well. Higher up and multi-pitch are Bell Canyon and Lone Pine Peak--both granite.
North of Salt Lake, just south of the Wyoming border (2 hour ?) drive are the Uinta Mountains with quartzite sport and trad climbs in a beautiful alpine environment at 10,500 ft. but very easy walks from the road. Maybe a bit chilly in Sept. but still likely climbable. Northwest of Salt Lake, in south central Idaho--3-4 hour drive, is the City of Rocks/Castle Rock area---one of the best, in my opinion, cragging areas in the US, in a truly 'wild west' environment--sport and trad and Sept is perfect.
Further south, in central Utah, again 3-4 hour drive is the extensive cobble-climbing of Maple Canyon---Riglos without the commitment. Joe's Valley bouldering is also in central Utah. Both these areas should be fine in September. Others have mentioned the pluses and minuses of the Moab/Indian Creek area. It should be emphasized that this area is well worth the visit even if you don't climb there.
Further south places such as Zion and the numerous areas around St. George are best reached by flying into Las Vegas. These areas will be hot in Sept., but shade can be found--and, given the low humidity of the desert, shady climbing is usually possible even during the summer (though currently with temps over 110F in that region that likely isn't the case).
Loads of other areas but this should give you some ideas.
