Living here, I answer questions like this all the time and I can not stress these things enough. Everyone thinks they are "use to hot weather" until they get here.
June is hot. I mean like 2 degrees cooler than walking on the sun hot. Early June is tough because our afternoon T-storms haven't quite kicked in yet so it stays hot and humid all day/night. By the last two weeks of June, the T-storms should be well into play. So plan on an afternoon T-storm between 2-5 everyday for about 20-30 min. Some people bring ponchos.
So first tip: DRINK WATER! like twice of what you think you should.
Take a break in the afternoon. If you are staying close by or on property, if you can, take a break from the parks in the afternoon from like 2-6. Go back in the evening when it cools down.
Use sunscreen and lip balm. Don't forget to reapply it cause you will sweat it all off.
Do 2 outside things and then an inside thing. Plan shows and inside longer attractions so that you get a break from the heat.
Oh, and DRINK WATER.
and don't forget to eat. Some people don't eat enough when they are hot which can be just as bad as not drinking enough.
If you can as well, bring something to wipe your face/arms down with. It will help keep you some what dry and cool you off a bit.
