While it's true that riding horses can be dangerous and does carry some risks, it is still a relatively safe sport as long as you have the right safety equipment and some good information.
First of all, nothing can reduce the risks of horsin' around as much as proper professional guidance. For beginners, it is crucial to seek some reputable instruction to get you started right. I've heard many "I rode a horse... once!" stories and they usually start out with the line, "a friend of mine had some horses and..." This story often ends badly unless your friend is a certified instructor or is a well known equine professional in your community with wonderful references.
However, once you've found a good instructor and you've signed up for your first lesson there are a few pieces of safety equipment that are crucial.
#1) A HELMET. This is an absolute must for beginners. I will not allow a minor to mount up without one. Most of my adult clients chose to use one as well. When we're starting out, even when riding the slowest, sweetest, most well trained lesson horse in the barn you just never know what uncontrollable variable might come up so be sure and wear a helmet.
Most beginner riding services can lend you one as you get started. If you plan to continue your equine education it's a good idea to purchase your own so you have a proper fit each and every time. Also be sure and purchase a helmet specific to riding horses. They are designed to protect you from the most typical types of falls or accidents that happen when riding. Horses are very different from skate boards and bicycles after all.
#2) BOOTS with a HEEL. The next significant danger you can protect yourself from is having your foot push through a stirrup when riding a horse. Boots with a heel of some type will do just that. While I use safety stirrups that also prevent this wherever possible I still encourage folks to consider what they are wearing on their feet. Again, for your first lesson it's not necessarily to go out and buy brand new riding boots. You probably have some low tread hiking boots or rain boots that'll do the trick for now. Running shoes are definitely NOT adequate so before you head to the ranch for your lesson, consider what's on your feet.
#3) OTHER HELPFUL EQUIPMENT
Long pants - shorts expose the skin to pinching and rubbing on portions of the saddle and other equipment and can be very uncomfortable. Gloves can be a good idea if you find your fingers are getting rubbed or you are struggling to maintain grip. Half Chaps or tall boots if you are riding in an English saddle.