Last year I went to Michigan Adventure as well as Disney World and observed people in all different kinds of shoes. Even a well planned out trip through the park puts miles on your feet, knees, hips and back. I know how temping it is to slide on those flip flops for a day in the sun, so many people I saw waiting in line for rides were sporting them. They look cute and show off your pedicure but by the end of the day your feet will be hurting. Instead, choose a nice vented pair of supportive running/walking shoes or sandals with an arch built in.
Drink plenty of water! This will help prevent your legs from swelling while you are on them all day. Depending on the amusement park, you may be able to carry water in a small back pack, others you may have to hand your water to a person not going on the ride with you.
Wear Sunscreen, even on your feet, if the skin is exposed then it can get burned. Nothing is worse than a sunburn on your feet and having to put your feet into a pair of shoes the next day.
If you are going to the water park, wear supportive sandals. This was the most memorable part of my trip to Michigan Adventure. They weather was perfect, not too hot, a little breezy at times. The reason it was so memorable was actually because my feet hurt, badly. I walked around barefoot.