I have a clear rain coat similar to the ones worn by the CMs, comes almost to the ground on me, but I'm short. Love it, I never get wet. If there is just going to be light showers, I take an umbrella but for the gully washers, the rain coat is my go to, it even has a drawstring hood. As far as shoes, I've never been when it's chilly so it's fisherman sandals for me. I would suggest taking a comfortable light weight tennis shoe made of mesh and throw a spare pair of socks in the bag if you carry one or your pocket if you don't. The shoes will dry quickly and changing socks mid day will help. Take at least 2 pairs of shoes, and I mean at least. When I used to take tennis shoes, I always took 3 pairs so the wet pair would have plenty of time to dry. I would let the kid walk, nothing worse than a soaked stroller for the little one. Mine was out of the stroller at 3. A child's poncho should be fine for the little one, they tend to drag the ground and they don't do a lot of moving the arms like adults do . If you use ponchos get a decent one that snaps and is sturdy, the cheap ones tend to rip and stick to you. Don't wear jeans, you will get wet and never dry out.
Only you know your capacity and that of your child, but I would be prepared to change your mind about three hours into the park experience. With any luck, they'll still have strollers available to rent at the front of the park.
