Our days are typically split into two halves: a pavement-pounding morning with a rope drop arrival, and a more relaxed evening with a sit-down dinner and a more meandering touring plan. As a result, we like to wear good sneakers in the morning (typically running shoes rather than cross-trainers or walking shoes, because they offer better cushioning and breathability, and with moisture-wicking ankle socks). After our afternoon break, we change into comfy sandals (hiking sandals like Tevas) for the evening. Changing shoes mid-day also helps keep our feet from getting blisters.
The exception to this rule is if we visit the Animal Kingdom and plan to ride Kali River Rapids, or if the forecast calls for a downpour: in that case, we skip the sneakers and wear our Tevas! I particularly recommend a hiking-style sandal in those situations because, unlike flip-flops or Crocs, which become dangerously slick and slippery when wet, hiking sandals offer stability and support in all conditions, and dry out in no time. Also, because they tend to be fully adjustable in just about every dimension, you can fit them to your feet and avoid any chafing that might lead to a blister. Knock on wood, using this little "system," no one in my family has ever had a blister or other "foot issue" on any of our trips to Disney World.
Whatever you decide to wear, be sure to break it in well before you go!