Just get the obvious and start using it with a note pad handy so you can write down items as you think of them. A few you may not think of though are:

fly swatter, heavy duty extension cord, chemicals for holding tank, flashlight, fire starters and a way to light it, margarita mix :), just seeing if you are paying attention. 

 In addition to other ideas above, I like the wrangler hay bag, rather than hay net.  Rubber boots!  Boot jack!  Membership in US Rider.  Tire air pressure gauge.  Mosquito repellent, patio mat, small whisk broom and dust pan (try as we might, we still track stuff into the LQ)  Banamine and bute paste.  Perhaps a spare used horseshoe, would be easy for farrier to reapply in a "lost shoe" emergency.  A roll of vet wrap.  A package of cable zip ties, trash bags.  Deflappers and tie down for awning.  Folding chairs and small folding table or two. Water hoses for trailer fresh water input.  two or three of those hose Y's that let you hook up 2 hoses to one faucet. 

 Your prescriptions in their original bottles.  Lip balm, ibuprofen or similar.  DVD player and movies, if it rains.  Depending on your eating situation, easy precooked possible things, microwavable snacks.  Headlight style flashlight (frees your hands for late night horse checks or untacking etc), glow sticks.  Info sheet for horse, as in who you would use as vet, any mannerisms or conditions (if you were unconscious, they might need to know he was head shy or did not tie well etc) Info sheet for you, emergency info, contact info, allergies, list of meds. Those sheets in a ziplock bag in your saddle bag and jeans could really help in what we hope is an emergency you never have!  Those reflective triangle traffic things to set by the trailer in case you have to be stopped on the shoulder.

Lots of great advice above-- US Rider is a Must!!

What we did is keep a notepad in the LQ and if it was something we forgot-- or didn't have ( our first season was last year) I would write it down so I would have it for the next trip.

I bought lots of the kitchen items from places like- Everything's a dollar- Big Lots etc and have my kitchen fully (well almost) stocked and just keep everything in there year round.

Have fun!!!!!

Maybe I missed it, but are you asking for what to take to shows or to take camping?  I know there's plenty of overlap but there are several things that are unique to each (I don't go to shows but I can't believe there's a big need for firewood ;) 

 If you're talking camping, is it developed or rustic?  Will you need some type of wheelbarrow, is it provided, are you supposed to scatter manure / hay, deposit it somewhere or take it with you?  Will you need your own picket line and associated items or do they provide it, corrals, stalls? 

Cooking style - entirely in the trailer, over the open fire, or somewhere in between?  We're discovering the magic and joys of the dutch oven but maybe you are happy microwaving leftovers. 

I've often thought a folding step stool would be really handy at the moment, but I never seem to remember to pack one, always make do with upside down buckets. 

And I vote for hay bags, Tough 1 I think is the brand I've had for about a decade now.

I'd like to add extra drinking water hoses(s) and a 30 amp extension cord. Also handy is a folding TV table. I find it's great around the campfire for snacks/drinks/marshmallows, etc.

You've had good replies to I won't repeat what people have said, but just add my two-bits.

Tack room - don't forget water buckets and a method to fasten them to the trailer. Hay bags are better than hay nets. Extra halter and rope for each horse in case something gets broke. Lunge line, lunge whip. All your tack including saddle, various blankets (I bring extras in case of problems) headstalls. A rain blanket or such in case it rains at night and your horse needs protection (most people don't have their horses sleep in the trailer). A horse first aid kit which includes bute, bandages and contact numbers for vet. A stepping stool/mounting block (comes in handy for many things, including saddling real tall horses!) your feed (You might carry it in the front stall) which includes hay, alfalfa cubes, grain, etc. Any suppliments your horse, and always take loose salt (I add to grain to encourage them to keep drinking) Twitch (in case of health emergency), hoof clippers in case of problem. Multi-purpose tool which includes a wire cutter. Fork, push broom for the back of the trailer, scoop shovel, wheelbarrel. You should be able to carry all this in your tackroom.

Yourself: All your personal clothes and items, including extras in case you get wet. Rain coat, gloves, head covering, helmet in case of bad weather. A big variety of clothing so you can layer. Warm pajamas at night so you don't need to run the heater too much. Toiletries such as soap, shampoo, toothpaste, etc. Pain killers, Gravol, cough/cold meds, antihistimines, any prescription meds, eye drops. Personal first aid kit.

Trailer: Stick to paper plates and disposable silverware and cups if possible, saves on dishwashing. Dishwashing soap, towels and clothes. Oven mitts. Fresh water (try to pack enough for your horse in case of emergency, too) Clean bucket to carry water for yourself if necessary. Basic appliances such as toaster and kettle. Cooking pots, bowls, utensils as needed for what you'll eat. Snacks to pack on the trail. Flyswatter, dustpan and broom. Throw rug at door so you can shake out dust instead of sweeping it all the time. Extra bedding in case it gets mighty cold. Matches, axe, flashlight, cell phone, emergency contact numbers. Alarm clock. Indoor/outdoor rug to put outside at door. Lawn chairs, fold up table. Kleenex, paper towel, toilet paper, chemical for toilet. Cleaning spray.

And money and a credit card! Oh, extra set of keys for truck and trailer. Good luck and have fun!  

I am still learning too. But I keep my trailer stocked, packed and ready to go with those essentials. You can find horse-themed RV mats on Ebay that are great. They can be easily swept off and stay clean. Extra hoses are good, extension cords, adapters. You might stock up with some canned goods too and non-perishables. I bought a lot of my stuff at Dollar Tree and Dollar General. I have a "house" broom and a tack room broom (to sweep shavings/poop) out of the horse part. I keep a pad of paper in mine and whenever I am running out of something, I put it on a list. I also made a list after my first few camping trips on other things I needed. It's kind of like playing house! You are going to have so much fun!!!!