1 - Water, even if you are not thirsty, make a point of drinking a few gulp often.
2 - Cooling towel. They really work! You put them around your neck, yes.
3 - Mist fan + spray. Dollar store have them, but of course you can buy a mickey one in the parks. We sprayed ourselves all the time.
4 - My daughter had a fan in her stroller
5 - Plan for a break or indoor shows in the afternoon. Carroussel of progress, Mickey Philarmagic, anything that is seated and indoor will help!
6 - "Quick dry" clothes, Usually says something like "dry fit" or "dry more". They are usually in the sport section. I wore polyester/nylon sport capri pants everyday. I had some other kind of pants that I thought were confortable in the heat. Boy was I wrong. I went to the closest WalMart and got more sport pants and bras! Seems dumb but a quick dry bras is so much more confortable!
I don't wear shorts or dress, but that would probably work too if you can prevent the rub.