We have state parks here in Texas. We can look up each park and find out if they have cabins available to rent, hook ups for campers or tent sites. Most of the parks have maps online, so we can pre-determine what spot we want (close to the bathroom facilities, close to the start of hiking trails, close to the river, as far away from the entrance as possible, etc). They recommend reservations but we went once where we just drove up and they have us a tent site. Once you enter, they give you a pass to put on your car w/your camping info (the number of yoru site and how long you'll be staying).
Most of our camping parks have 24hr security, where some sort of patrol drives around where the campers are.
I like tent camping near water and electric hook ups. I love roughing it but w/the option of modern amenities if I want them. I am a minimalist when it comes to camping - my tent, a sleeping bag, bottles of water and granola bars for my meals. But since I have a son and my husband isn't as much a minimalist as I am, we bring a portable stove, air mattress and a cooler of ice for refridgerated stuff. I limit our cooking gear to 1 pan, 1 spatula and we eat off paper goods.
Most camp sites have hiking, fishing, parks to play - whatever you like to do w/your family. People usually sit around and snack on junk food or s'mores, the kids ride their bikes or go exploring or play in the dirt or swimming, and at night we make a fire and eat more s'mores. So decide what you want to do and find a campsite that offers those activities.
Camping is so much fun w/kids. Don't be scared!