If I don't expect rain, I usually keep a disposable poncho in my back pocket. You never know in Florida.
If I DO expect rain, I go ahead and bring a lightweight rain jacket. Ponchos are handy in a pinch but pathetically flimsy and uncomfortable in actual practice. There's no real reason at Disney's parks not to have at least one person in your party carry a backpack, and you can stuff several rain jackets into one. Otherwise, tie it around your waist when not wearing it. Like
@The Mom said, having a cap with a bill will make any rain gear with a hood work a lot better.
Sandals with nylon straps and rubber footbeds, like chacos or tevas, are ideal for rainy weather, since you can walk through puddles and they will immediately start drying out.
At theme parks I usually keep at least one zip lock bag in my pocket for my phone or wallet, both for water rides and possible rain.
If possible, shorts made of synthetic, quick-drying material are better than cotton. Ditto with underwear- if you anticipate getting wet, something tighter and made of synthetics will dry sooner and prevent chafing. Underarmour athletic stuff, or anything similar will be way more comfortable than cotton.