S.C.
We camp quite a bit and have a 2 year old. Here are essentials I've come to count on: Stove with fuel canisters, air mattress for tent, food pavillion for coverage in case of rain (and a place to retreat to in case of rain), games to play with everyone, fun snacks for kids that they wouldn't usually get to have, flashlights for everyone (fun for kids too), hatchet for chopping wood for fire, lantern, blanket for kids to sit / play on, camping chairs for everyone, marshmallow roasting sticks, rain gear for everyone, extra clothes for everyone, a bin for washing dishes and washing kids at night, water, cast iron skillet, cooking utensils, trash bags, towel / washcloth for washing up, towel / cloth for washing dishes.
for food: grilled cheese, brats, hot dogs, hamburgers, soup (get a starter mix that you just have to add water to), cheese and crackers, fruit, jiffy pop popcorn (fun!), smores, pancake mix that just takes water, egg beaters (easier than eggs). also, if you have time, you can get stuff chopped / marinated prior to camping, so you don't have to do any prep at the campsite.
Just some ideas...