Kaylypso, Zion isn't that far away, but the climbing areas will be very hot.
I would highly recommend that you visit the cool slot canyons instead of trying to go rock climbing.
Bull Valley Gorge and Round Valley Draw are two good ones (non-technical) that are nearby and the shaded canyons will be farily cool in the summer. Some photos of some of the nearby slot canyons are below:
http://www.zionnational-park.com/stairc ... -gorge.htm
Escalante area is also nearby and has a lot of fantastic things to see in summer:
That time of year (unless it is late September), I would suggest going those places rather teh classic rock climbing routes in the desert. Many of the slots themselves are rock climbs (especially around Escalante), but the ones around Tropic aren't guided (as far as I know).