My family and I were at the World for the first two weeks of this past June and we found it to be a pretty good time to be there. As a matter of fact, we have two trips scheduled around that time this year - the last week in May and the third week in June. We were actually planning a two week trip but a co-worker needed that week, so I moved my second week till later.
I think you will find the weather warm (80's to 90's) but it will not have reached the much more uncomfortable 90~100° temps of july and august. Like many before have said, hit the parks in the morning, take a break mid-afternoon at the resort, resort pool, or water park, and come back to the park in the evening cool and refreshed.
There are some definite advantages to this plan. We found that by afternoon the parks were their most crowded, and alot of the "park commandos" were getting somewhat cranky, along with their tired and hot children. At this same time many of the persons who crowded into the water parks when they first opened in the morning were leaving. You may draw you own conclusion on what to do with the information above

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We used this several times on our trip. Grab a FP for Pooh with a four hour return - no problem ! A four hour return in this case, allowing for transportation time, means two hours swimming at Blizzard Beach !
Let's see - what else ? Take advantage of FastPasses (of course). Use a combination of inside and outside rides to give you cool periods during the day. For hot days, bring misters with you, and put ice in them before you leave the room. If eating in the MK, remember the second floor of Columbia Harbor House - most don't ! Same for the bathrooms there - you will often see lines for the first floor bathrooms there and empty stalls at the second floor ones. Plus, it's really neat to eat in the area that extends over the walkway between FantasyLand and AdventureLand - a great place to people watch in the cool

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Most of all - Have Fun ! You can't see everything, and you can always come back.