I can rattle off a list of all the stuff that I think is super-important to pack when you're camping, other than the obvious like your tent, and I will do that, but the thing I want to point out with your camping supply list is two things.
First, it will evolve. The more camping you do the better you will get at knowing what you need to make your life better and make you happy out there in the woods. The other thing I really want to stress is you should type up a camping list and you should live by it religiously.
You should keep adding to it every time you go on a trip and you come back. You should add to it the stuff that you forgot and wished you had. You should take the stuff off that you brought and you never even touched. A camping list is something to really live by and it's something to get better at making and so you want to have it typed up somewhere so you can always revisit it.
So what's on my camping list? Let's see. Duct tape because things break and duct tape is really easy to use. If you have a hole in your tarp you can use duct tape. If you get a hole in your sneaker you can use duct tape. Rope, you absolutely need some kind of rope to hang food, to do lots of other stuff that you'll be surprised about, to hang your tarp, to set up a clothes line, dry your clothes. Toilet paper, paper towels, the things about those is that you should take the cardboard out of the center, you can squish the toilet paper or the paper towel and work the cardboard out. You can also bring the cardboard and use it in your fire.
Speaking of fire, you definitely need matches and lighters and you need lots of both. And the thing I like to do is stash them all over the place. So I've got a couple in the pockets of my backpack, I've got a couple in my jacket. You always need them and you usually never have enough.
You're going to need a tarp because you're going to want a place to be other than your tent in case it rains. And a tarp is just good to have. You could even use a tarp to put all your food in, wrap it up on one side and hang it. You could even use a tarp to collect water if you had to. A tarp is just a super-handy thing.
You're going to need something to cook with. You can always use the fire, but in case you're having really bad fire luck, which happens, or in case it's raining, you should bring a camping stove, some form of camping stove. I have a really cheap one. It's just a burner that attaches to a butane tank and it does the trick and it's really light.
You're going to need pots and pans. I just really live with one pot and I use that pot to boil water and to make soup, and then one fry pan and I use that for eggs and most of my other meals. You're going to need a knife, a really good knife to chop up stuff for dinner.
Also, you're going to need a knife just to cut the rope that you're going to use, so a handy pocketknife is definitely good to have. Some sort of hatchet or a camp saw is good because although you shouldn't be cutting branches off of trees, you may find a piece of wood that's really great for your fire, but it's huge. So you can chop up firewood that way and make it into more manageable pieces.