In very broad general terms, and YOU will need to select the specifics:
1. Personal hygiene items, everything from the tooth paste to the Q-tips to the hair spray, bath towels to toe-nail clippers.
2. Food preparation items. This actually falls into a couple categories.
A. Inside cooking
B. Outside cooking
Are you planning on cooking on the camper stove inside? Then you need to consider your pots and pans. Are you cooking outside? Then you need to consider what will be your heat source? Electric griddles, BBQ's, fire pit, Coleman Camp stove, crock pot, Dutch Oven? Then have the utensils to make that happen. We have pots and pans that will work over a fire pit and on the trailer stove top, but we also have 2 electric griddles (which we use 99 percent of the time) outside. So that also means extra extension cords.
3. Some kind of first aid kit, at least some extra bandaids.
4. Outside lighting for after dark ... consider both battery powered and AC powered lighting... Flashlight to awning tackie-lights. Still, something to light the dark.
5. Your food, your toys, your hobbies, (including television watching, computers, I-pads, cell phones, any other type of electronics for entertainment), fishing pole, magazines.
6. Dinner wear. Dishes to eat off of, if that be plastic, paper, or Grandma's old china, you need to have something to eat off of and with (forks, spoons, plastic or metal)
7. Trailer set-up.
A. Something to level the camper. The tires should be on some kind of leveling blocks, if that be wooden boards or Lego-style pads. You need something to level under the tires and under your stabilizer jacks and tongue jack.
B. Something to plug the camper into electricity with (an RV extension cord... 30 amp or 50 amp, depending on your camper.) 50 foot or 25 feet lengths, whatever you may think you'll need. We carry enough for about 175 feet.
C. Water supply? Garden hoses. Usually white is designated for drink water, but some of us old timers refuse to give up that old green hose laying in our yard. Your choice. Take your own chances with how strong your immune system has become. Mine's pretty good after 63 years.
D. Sewer (black and grey tank) hoses. Here again, there are expensive ones and cheep ones. Your choice. We've had both, and they both eventually fail. Length? That's a matter of preference. I like to make sure I have at least enough to reach from the sewer outlet on my camper to the farthest end of the camper and about 10 feet beyond that. Sometimes, weird campgrounds will have the sewer drains at the rear of the campsite pad. I carry about 100 feet of hose all the time. Usually use only 30 feet of it though, because of the location of my drain on the camper.
D. Chocks! Most important. Chocks. Get chocks and use them every time you unhitch the camper, even in storage, and even on flat ground. NEVER, NEVER, NEVER forget to chock, even if you think it's not necessary. Do it! There are several different kinds and styles of chocks. But even a 2x4 can be used as a chock.
E. Outside furniture (folding chairs, collapsible hammocks, outside kitchen set-ups, folding end tables, substitute picnic table in the event you are ever somewhere that might NOT have a picnic table. Outside ground mat (your choice), outside lighting.
F. Bedding? Sheets, blankets, sleeping bags, pillows, electric blankets? (oh yea, really nice for Spring, Fall, and Winter camping). Something for every bed in the camper, and tailored to each person's likes and dislikes who will be using that bed space.
8. Odd's and ends: Like extension cords for any appliance you may have with you. A small tool set, including a hatchet and a utility knife. Here again, take only tools that are needed for the camper and only what you are able to fix on the spot. More is not necessary. Items like duct tape, electrical tape, bungie cords, rope, cloths pins, string, extra batteries for any battery operated device you bring along. Are you planning on bringing your own firewood? You may want a tarp to cover the fire wood and some way to pin it to the ground (like tent stakes). Do you have pets? Dog dishes, leashes, pet food, dog shampoo? It's all up to you.
Basically, anything you use in your house, you will most likely use in your camper too. Think of your camper as an extension of your house, and stock it that way. Only you know what your likes and dislikes are, your style of living, and how much you can afford to spend, beg, or borrow to stock the new camper.
But, once it get inside the camper, keep it there. Get a second one to replace the one in the house, or buy new for the camper. And, by the way ... you will always, and forever going forward, be stocking your camper. Three years from now, you'll throw out all the unused "stuff" and 2 years later, you'll be doing it again. It never ends. It's important to control your compulsive instincts to want to razzle and dazzle everyone with your great accumulation of camping gadgets. It's not necessary. And eventually, you'll find there is really nothing special about camping gear and stuff in your house, except maybe where it's used.
Good luck! Enjoy! And congrats on the new camper!