I spent a good portion of my young life outdoors, so take this with a grain of salt, but when travelling with the kids. We camp on the way in state etc parks ,( private campgrounds are getting 40 bucks a night for a tent spot in some places now, parks are 12-18 usually. Although did see 24 in Colorado) ..we just pack tent, sleeping bags, ground pads a single burner backpack stove , easy grub and whatever essentials.most modern vehicles don't have the carry capacity of a 62 mercury wagon, so if you haul too much. It can be a nightmare. We fit everything for 4 people and a large dog for 2 weeks nside a volvo wagon, but its tight. In that climate, treat the tent rainfly and lower exposed surfaces with water repellant or you will wake up wet.

I use a Coleman propane stove. You can get your pots and pans from your kitchen, but they might get beat up a little.

Rather than go throughall of it, here's an REI site list.
Their list is too inclusuve, but it's a good place to start.

[rei.com]

State parks are also nice because most families are friendly, and everyone love to show off their camping skills. You may be offered more help than you're ready for.

As suggested, state parks are the safest (running hot/cold showers, store, some even have laundry)

Looking for safe but rustic. National forest campground. Vault toilets. Safe drinking water. Trees and wildlife.

If you are looking to backpack, you might want to do a few parks first to get a feel but stay a few feet from your car.

State Parks are almost always a good bet, especially if you don’t have a great deal of experience. As already mentioned, an air mattress. Doesn’t hurt to have a couple of folding chairs. A main issue is what you're going to do for food. If you plan to be self-contained (no driving to restaurants) you’ll need a campstove. A two burner propane stove is nice; single burner could suffice. A cooler for iced food storage. Plenty of water. Some sites have water and electricity; some don’t. Adequate hiking shoes, already broken in. A first aid kit. Toiletries. Rain jacket. Lanterns for the tent and/or picnic table area, preferably battery operated. Small pocket flashlight. Lighter for campfire. Firewood, bought or gathered. Maybe a hatchet for splitting kindling. Insect repellant. Phone charger. Pocket knife. Some cordage for a clothesline for drying damp socks/whatev. or for hanging a lantern.

An easy to pump up air mattress and day backpack. Head torch. Water bottles and snacks. Suitable clothing for the weather. That’s pretty much for the essentials, Depending on what you plan when you get there and the provisions situation/ cafes.