I live in FL and go to the parks a lot when it is hot. I've soaked shirts from top to bottom mowing the grass in FL, so have tried a few things.
Polo - some guys like to wear polo shirts, and I will wear them on occasion to the parks. There are (2) kinds I recommend - the Nike Dri-Fit or pima cotton style shirts. They are both light weight, repel sweat, and dry quickly if they get wet from sweat or an afternoon shower.
T-Shirts - can't go wrong with a nice loose fitting cotton t-shirt. They too will dry fast if they get wet. However I really like the polyester sport/workout style shirts. Thye are super light weight, repel sweat, and dry super fast. You can buy name brand, get you favorite sports team, or pick up a cheap one at Wal-Mart for any size budget.