People here have long lists of gear. A lot of that stuff is nice to have, but you don't need much starting out.
Here's what I use:
Tent - I have an old, cheap one that has worked great in mild weather for the last 15-20 years. If it's going to rain I prefer a tent with a full coverage waterproof fly and vestibule area to keep shoes dry (REI Half Dome style) but it's not necessary. I've strung up a tarp over the old tent on several occasions, including one super heavy thunderstorm. Keeping the walls of the tent dry (to reduce condensation inside) and making a small dry area to enter/exit the tent and put on your shoes and rain gear is really nice. If the places you camp have trees you can string up a cheap blue tarp with Paracord and stake the corners to the ground to make an A frame.
Sleeping pad - I used closed cell foam ones for a long time, but I could never sleep well unless the ground underneath was very soft. I tend to sleep on my side so the side on the ground would go numb, making me wake up and roll over. I have an inflatable pad now and it's a huge improvement. I've used it more than 30 nights with no holes yet, and if it leaks I will patch it. Two foam pads is almost as good as an air pad, and probably cheaper.
Sleeping bag - Buy one rated for about 10 degrees colder than the nighttime lows you are expecting, or rated for even colder if you get cold a lot. If you're not carrying it in a backpack or camping in below freezing weather there's no need to spend a lot of money here. My Kelty 20 degree bag was great for about $50, and as warm as a 0 degree bag if I cinched it around my head, but I replaced it since it was too heavy/bulky for backpacking trips.
Light source - A headlamp is best, ideally one per person but you can share. Since it frees up your hands and always illuminates where you are looking, it no longer matters if it's night or day. Lanterns always throw off too much glare for me, and you can just put a headlamp in the top of the tent if you are inside.
Water container - I like to use a water bottle that won't leak so I can put it next to me in the tent.
Glasses or contacts case, if needed.
A pot for cooking. A pot can work as a frying pan, too. Stainless is best since it can tolerate overheating. I have a backpacking pot, but I've also used one from my kitchen on a grate over a fire. It's permanently black on the bottom now, but that improves heat transfer. Keep plastic handles out of the fire. Cast iron is ideal, too.
A spoon - This is often the only utensil I use. I almost always have a knife, but I don't always use it.
A bowl, if more than one person is eating (otherwise eat from the pot). I use a $0.99 bowl from Walmart that's not intended for camping. A tupperware works, too, and it has a lid so you can seal in the mess and clean it at home. Plates are not needed if you have a bowl.
Clothes appropriate for the weather.
Toilet paper, a small trowel (poop shovel), and hand sanitizer if facilities are not available where you camp.
Some general advice:
Make sure to always secure your food from animals. In the car is usually good enough unless you're in grizzly bear country (they will rip the car open).
Learn about sanitation in the woods if there are no facilities where you are going. The poop shovel should never touch poop!
A saw is more effective for cutting wood than a hatchet. Most designated campgrounds don't allow cutting down trees, but you can cut up fallen logs.
You can sit on logs instead of camp chairs.
I've found that not using toothpaste works just as well as using it, and missing a few doses of fluoride won't hurt you.
Instant oatmeal packets are small, expensive, and loaded with salt. I need to eat three or four (half a box) to make a good breakfast. Buy quick cooking oats and add spices, brown sugar, or fruit. You can mix in an egg to make it rich and creamy, but a serving of oatmeal already has as much protein as an egg.
If you're not backpacking you don't need expensive gear for camping.
Learn how to build a good fire, and bring some things for reliable fire starting. Dryer lint is good. If it's always wet you might want to bring dry wood from home.
Layering works with sleeping bags too. I have a 30 degree bag for summer, a 20 degree bag for spring/fall (often below freezing at night. I slept on snow earlier this month), and if I put the 30 inside the 20 it's good down to below zero if I cover my head and face enough. Sleeping pads are also an important part of your insulation since the sleeping bag under you is compressed and the ground is colder than you think it is. Get an insulated air pad or use closed cell foam pads. Or use foam pads under an air pad.