Start with a good cross country vest, obviously a good helmet, and good boots for horse and rider. A medical armband for shows, too.
Seriously, my list is MILES long for both schooling and events, but I'll give it a shot:
horse--saddle, pad, bridle, girth, braiding kit
rider--helmet, hair net, coat, shirt, stock tie, stock pin, gloves, breeches, belt, boots, dressage whip, spurs
horse--saddle, pad, bridle, girth, breastplate, running martingale, boots, bell boots, studs, stud kit
rider--helmet, shirt (many say polo, but I just use a T-shirt and NOBODY--including my coach, who cares about turnout--has ever commented), vest, armband, gloves, breeches, belt, boots, bat, spurs
horse--saddle, pad, bridle, girth, breastplate, running martingale, braiding kit, boots
rider--helmet, hair net, coat, shirt, stock tie, stock pin, gloves, medical armband, breeches, belt, boots, bat, spurs
Remember that at a show one usually rides the night before, so schooling pad, breeches, etc. are also needed
Stabling--buckets, feed pans, stall guard, pitch fork, wheelbarrow, broom, hose (not always needed), hooks, snaps, hay net, blankets
Grooming--brushes, hoof pick, fly spray, show sheen, any other show grooming materials, wash bucket, scraper, shampoo, conditioner, liniments (show legal only!), hoof polish, scissors, small clippers, towels, rags
Post Ride Care--ice boots, poultice, paper, standing wraps, any magnets or lasers or other therapy you use, cooler, epsom salt
Gear care--tack sponges, saddle soap, conditioner, boot polish and brushes, rags
lunge line/whip, halter and lead, farrier tools, spare buckets (always mark drinking water buckets vs. other stuff buckets), horse first aid kit, stud chain, saddle rack, bridle hooks
hammer and nails, WD 40, extension cord, tape (duct, electrical, masking), baling twine, pens, paper, sharpies, human first aid kit, bike, good wellies or muck boots for walking through the water jump, wheel for course walk, shipping boots
I also bring camping chairs, cooler, tent, camping grill, folding table, sleeping bag, pillow, book, food, and clothes of course. These are not horsey items, but things I need for going to shows!
I am obsessive, so I bring spares of tons of things.
I am sure I have left off lots of things, especially considering I haven't been to a show in close to 2 years.