Don't EVER walk under or near the back of a horse - you could get kicked or bitten and you can die
Don't yell OR run around a horse-he/she could get spooked
Don't wear shorts if you fall you could get scraped vey badly
Don't tie a horse by its reins. It the horse panics and tries to pull away, the reins can twist around the neck and cause the horse to be suffocating, fall over and so on.
Do wear a helmet - when falling from a horse, a head injury is the main reason of death or serious injury. Some riders choose to not wear helmets because they don't see the risk - it is highly reccommened, and most riding schools will not accept a rider who refuses to wear a helmet. Even the most experienced riders should wear a helmet, but it is one's personal choice. Also, it is a must to wear a helmet which is not damaged and certified!
Do double-check your tack to be safe. This is a must. When mounted, it is safest to get someone to check it from the ground, but it is also safe to check in while mounted.
Do mount from the left side of the horse. Only experienced riders should attempt from the right side. It's very unsafe, so please don't attempt this.
Do wear heeled boots/shoes. This does not include heeled decorative boots. The only boots that should be worn are tall boots and riding boots. Work boots are not built the same and could cause pain to the horse. If boots aren't avaliable, wear sneakers ONLY, but this is highly frowned upon.
Do wear jeans or jodhpurs. If you fall, most items of clothing won't protect you, however, jeans and jodhpurs do.
Do use mandatory tack! It is extremely dangerous to ride without a girth when using a saddle and is not at all reccommended, even to the most experienced riders out there.