Earlier in the morning is the best way to beat the crowds. We found the Early Entry or whatever it is called now for MK was good to get Fantasyland done with little ones.
Take advantage of Fast Pass. It really helps. For instance, get Fast Pass for Splash Mountain, then go do other things that are shorter lines like Pirates, the Railroad, etc. while waiting. If things have not changed from last year, you are not committed to go that exact hour, you must go after that time. For instance, if you have a return for 10am-11am. You can go ANYTIME after 10am which could be noon. (correct me if they changed this since last June)
Take a break in the afternoon if you can at the resort for a swim and rest. It is the hottest time and the pool helps. Speaking of hot, it is HOT HOT HOT! Being from a dry climate, I find the humidity makes it feel hotter for me. We bring Squeeze Breeze or misty fans to help keep cool.
Bring ponchos in your belongings. The weather starts to bring in afternoon rain clouds at that time and sometimes it rains. It seems that whenever I forget the ponchos, it will rain that afternoon! The rain does not normally cause any problems and we just keep going.
We will be there also during this time. This is our third June trip and we enjoy this time. There are still schools in session, so the crowds are not as bad...they are managable. We are doing the USO/IOA/Sea World and WDW water parks that week you are there. We are heading into WDW our second week. Our suitcases are out and begging to be packed!
Have a great trip! Don't forget to wear a lime green ribbon somewhere to identify yourself that your are a DISer; you may meet a new friend there!!!!!!