We have been camping for over 20 years now including a trip cross country and back which made us into semi-minimalist campers - not a lot of extras BUT there are some things you just gotta have and most can be had at the dollar store or Walmart or Target or YARD SALES
. A few things I recommend getting at a camping store like BassPro or Campmor.
Two good rubbermaid tubs to store all your gear in. We have the "camping" stuff in one and the cooking stuff in the other so the dishes don't end up with dirt from the tarps, etc.
Stove & a Lantern (or 2) - definitely want Coleman. They can be had at Target or Walmart. Used can be had cheap on Craigslist or yardsales (or even better - dumpster diving - I picked up our stove and lanterns on the side of the road - someone had put them out for trash - they just need a good cleaning). Despite plans for cooking over the campfire you really will want to get a stove as a backup. All it takes is a month without rain for campfires to get banned - some places never allow them - and cooking mac n cheese over the fire is a PITA. No need to go fancy - two burner stove simple wind shield. For a latern - I wouldn't do battery for this the batteries just don't last long enough. Propane or dual fuel are best. Always buy a set of extra mantles to keep in the box. Definitely don't be afraid of old. The lanterns and stove I found are from the 70's and our secondary stove is really vintage (as in 1930's - out of my grandma's basement) and they all still work great. The vintage are actually better than the new ones we have as they are made in USA and are bigger and better/heavier metal was used.
A dish pan - most campgrounds REALLY frown on you washing dishes in the bathroom sinks and not all of them offer up a wash station.
I second or third hitting goodwill for plates, cups, utensils, pots & pans. I got all our camping silverware there for under $2. I found that vintage tupperware can be had on e-bay for cheap and is really great cause it nests. Unless you decide to get a JetBoil or Don't waste your money on special "camping" cooksets. The quality just isn't there for the price they charge.
Plastic tablecloth. Trust me. You'll want one.
Don't get the one in the camping aisle. Hit the dollar store for a flannel backed one and it will last you - we've had ours going on 10 years.
Tarps and bungie cords and some sturdy rope - You'll want to string one over the table. Enough rope for this plus for a clothes line for your towels.
A package of clothes pins.
A roll of quarters.
A good hammer with puller or camp mallet for when you camp in areas with rocky ground.
Dust pan and whisk broom - so we're not sleeping in dirt and/or pine needles.
LARGE plastic pitcher (with a lid) for holding water. If the water is really far away then you may want to invest in a 5 gallon jerry can designated for carrying water and keeping it at your campsite.
A french press works well for coffee and can be had for under $20 - when going really streamlined DH uses instant.
We keep a box of Extra Long Matches AND a candle lighter in our box. (The lighter is good for a campfire and the stove but too wide to fit in the openings in most lanterns.
Flashlights and/or headlamps for the kids.
Airmattress - I got a self inflating queen sized Coleman at BassPro for $35.
ETA - One item we wouldn't be without - Good long Metal Marmallow Sticks. Good for cooking hot dogs and shish kebab and I always set one to the side to use as a fire poker.
Good Luck!