We camp quite a bit with the kids and with a big family, packing light is crucial. Your link didn't open to the tent, btw. My first piece of advice would be to check the tent stakes before you camp. Generally, the ones that come with the tents are flimsy and bend easily. You can get a set of good ones for a little extra money from a sportings good store (or Amazon) and they'll be well worth the extra few bucks spent. Trust me on this one!
For mattress pads, something like this will serve you well. Takes up little space in your car, but they are essential if you're tent camping in a sleeping bag.
We've got a propane fueled camping stove that's sleek and does well. We bought a new one this summer and love it. The sides help shield wind. But honestly, I can go with or without the camping stove because I really like using our cast iron skillets over an open fire. I've got a reversible flat cast iron grill/griddle that might be one of my most treasured possessions. Get one of these and you can master the dojo around a campfire. Bacon, pancakes, burgers, salmon, eggs, steaks.....you name it, this thing will cook to perfection over an open campfire (most campsites will have a fire pit with a heavy grill grate there - if not, you can always straddle this over rocks). The reversible thing is key because my wife is a vegetarian, so I can cook up hamburgers on one side, flip and cook her veggie burgers on the other (for instance). If you're not familiar with cast iron, read just a little on seasoning them, cleaning them and caring for them. As old @cosjobs will tell you, you can clean your cast iron by cooking up some bacon and then just wiping it down with a paper towel and he's really not kidding.
If you like coffee in the morning, I suggest a French press for campers. We're coffee lovers, so making a nice pot of fresh coffee in the morning is key. One little camping hack that we do for this is to bring our tea kettle. Set that over your stove or campfire and you get boiling water in a hurry absent any sort of ash or debris landing in an open pot of water. Can also use this if you make kids hot chocolate.
I've got a camping box of supplies with things like heavy duty oven mitts for the cast iron (they get hot!), metal tongs, spatula, lighters, paper towels, can/bottle/wine openers, first aid kid, eco friendly dish soap, sponge, hammer (for the tent stakes) extra batteries for flashlights/lanterns, camping utensils and cups....I've camped for well over 10 years and just sort of created a little box that's ready to go at the drop of a hat. Not unlike an emergency preparedness kit. You'll discover what you need and probably more importantly what you don't need the more you go.
Make sure you have a good pocket knife - I've got a nice Swiss Army one that has a bunch of uses and I take it everywhere. The most important part - of course - is making good s'more sticks. Don't be a wimpy dad and buy the metal s'more rods. No, go out with your kids, find a good stick and then whittle it with them for the nightly s'mores and you WILL make these, right, you're not a psycho?
When my wife and I got married 10 years ago, we had one of our registries at REI and that was perfect for us. Somebody bought us a pair of their camping chairs and they are not only comfortable, but incredibly durable. You don't have to spend a ton of money on good camping chairs, but you get what you pay for with these and to me, there is NOTHING better than sitting out under the stars by a fire with an adult beverage resting comfortably in my camping chair drink holder.
Last suggestion: Get a camping hammock. We have one like this and it's the best. Find a couple of trees, rig this thing up and enjoy your time outdoors in comfort. Takes up NO room in the car, either. Great $25 purchase.