If you can, take a few "practice trips" beforehand - maybe just an overnight, not too far from home. That way, if you do forget something or think of something else you should have brought, it's not going to be a HUGE deal but you can write it down and remember to bring it along on your next trip.
Depending on the accommodations, we either tie our horses or rent stalls at the campground. Stalls are always nice for longer trips, especially since it always seems to rain on us and it's nice to know the horses are under shelter and can turn and lie down. In the spring and fall we bring rain sheets, especially if there are no stalls available.
Be sure to plan out how much food your horse will need for the duration of the trip, and pack a little extra just in case. I always bring a tub of Finish Line Ultrafire electrolyte powder to mix in with their grain or water, and a small tub of bute powder, along with a horse first-aid kit and book. The electrolyte is a good thing to start them on a day or two before you leave - add a little to their water or grain. Some horses don't like to drink "strange" water in a new place, so when you get to your destination you can add a bit of the electrolyte to the water there to encourage them to drink.
Once fly season hits, you'll want to bring a bottle of fly spray, maybe a fly bonnet or mask for the campground if your horse is really sensitive to bugs.
Don't forget things to take care of yourself - a first aid kid, some bug spray and sunblock for yourself. Not to mention the usual toiletries. If there is a shower house at the campground, you may want to bring a pair of flip-flops to wear in the shower.
If you're camping primitive, bring a flashlight for those "night checks" on the horses. Propane camp lanterns are great for lighting up the campsite, especially when camping primitive.
Also helpful are those little "snap" glow sticks or bracelets that can be attached to a horse's halter at night so you can see them more easily. They're easy to attach, easy to spot, and can often be bought in bulk at really cheap prices from places like Oriental Trading Company.
When it comes to food/eating, we pack a lunch in the saddle bags (I have a fondness for bagels with flavored cream cheese, and some kind of salami or pepperoni - yum), little snack bags of chips or Chex mix, maybe beef jerky or summer sausage.
Although I have quite a few friends who like to drink beer when they're out riding, I just stick to bottled water. Nothing carbonated or fizzy. I may get those little individual-sized packets of Crystal Light or other flavored powders that you pour right into the bottle.
Consider bringing comfortable (but safe) shoes for around the campsite, like slip-on muck boots in case the ground is wet or muddy.
For campsite eating, again, it depends on what type of camping you're doing. . .but I like a good meal after a day on the trail.
If we're at a campground with electricity, I'll throw boneless skinless chicken breasts in the crock pot with a jar of chunky salsa and let it cook low while we go out riding. When we get back, I warm up flour tortillas on the propane skillet, grill some green and red peppers and onions, throw on some chopped jalepeno peppers, and we have ourselves some chicken burritos.
We have a small propane grill with different attachments (a grill, a burner, a skillet, etc), though I really prefer open-fire cooking for things like steaks and burgers.
Other favorite meals are red/black beans and rice with sausage, or chili. Finish off the evening with a red Solo cup of blackberry wine while we sit around the campfire. . .yep, I LOVE horse-camping.