I've only done West Rim, also in June, with my son (who was 9 at the time), and a friend with his two boys.
We started out around 2pm at Lava Point and arrived at Potato Hollow Camp 8 at 5pm which was near a spring. That turned out to be a good thing because all the rest of the springs along the trail were dry. We ran into 3 hikers who had to backtrack to this spring to fill up. The next day makes for a long 9-11 mile trip and you will want to make sure you have enough water.
Photos here: http://flic.kr/s/aHsjkx1HQn.
Good description: http://www.protrails.com/trails/view/291
Here's a map: http://www.zionnational-park.com/map-west-rim