In Zion, perhaps the only real slot canyon that is easy to get to is Echo Canyon from the bottom. You can get there from Weeping Rock.
You can see the bottom end without ropes:http://www.summitpost.org/echo-canyon/300948
Also consider the Wire Pass near the Paria and some of the slots around Escalante (Egypt 3, Peekaboo, Spooky, and Brimstone Gulches), but get an early start in order to get into the slots before the heat sets in. If you get an early start, they should be somewhat reasonable in summer. Not far from Bryce are Bull Valley Gorge, Willis Creek and Round Valley Draw which are reasonable in summer. Strenuous, but not technical.
Upheaval Dome will be brutally hot this time of year if you are hiking. Personally, I would wait until October unless you just want to see it from the road.
Some really good areas in summer:
Bryce Canyon (a bit hot in the afternoons), Cedar Breaks and Calf Creek Falls (near Escalante). Around Moab (If you do go to Upheaval Dome), try the La Sal Mountains, Mill Creek, Negro Bill Canyon and Professor Creek. Most other places will be brutally hot.