If it's going to be a long day, I'll wear running shoes. Sometimes I've worn Teva mush flip-flops, Crocs (the regular kind with the back strap), or a pair of sport sandals (I forget which brand). I don't like running shoes if I'm going to get wet on rides or from rain. They are definitely the best thing for my feet otherwise, though.
I'm considering trying Crocs flip-flops (Capri style, I think it's called?) or my water shoes (they are amazingly comfortable, even when wet, but I've never tried them in an amusement park). I might try out one or the other on a morning and see how it goes.