Raingear: always have it with you. Ponchos or raincoats, yes, but also plastic bags for electronics and/or other fragile objects.
Always have sunscreen, sunglasses, maybe a hat. If you're fair-skinned like me you may wish to even bring a light jacket to help with the sun exposure.
Have a bookbag or bag with containers you can pack everything in quickly. Since you're traveling from place to place you want to have everything you might need with you, which includes first aid, water, snacks, sewing kit, extra batteries, your flashlight, etc. Wear good firm shoes you're comfortable standing in for long periods of time that you don't mind getting muddy, dirty, possibly wet. Have an extra pair of socks in your bag in case you need to change quickly (extra clothes in general are good).
Constantly check weather reports. Find out what your company's bad weather policy is: not just rain but also extreme heat, tornadoes, hurricanes, etc. Know who makes the decision on a cancellation or delay, when this decision needs to be made, and how to alert your audience. Find out if the locations you're going to have any special rules regarding guests (open containers, coolers, etc.) Know where your actors will be able to change or if you'll need to setup your own changing station. Be prepared for audiences to show up way early. Also find out what other activities might be scheduled at the park for the same day: political rallies, fairs, kids days, puppet shows, etc.