For safety reasons, you must wear an ASTM/SEI-approved helmet, boots or shoes with defined 1-inch heels, and long pants or breeches to participate in lessons. The club will provide helmets to you at no extra cost, although you are welcome to purchase your own.
Proper footwear is not only safer, but more comfortable and easier to ride in. Here are some examples of proper and improper footwear:
The boot pictured in the middle is an excellent choice for riding. It is a paddock boot designed for riding, and it has an appropriate heel height, stiffer sole, and will give the rider's foot more support in the stirrup. While the rider's foot can still slide forward in the stirrup, the boot design will likely keep the foot from becoming stuck in the stirrup, which is important if the rider should fall from his/her horse.
It's not necessary to purchase boots that are specifically made for riding, but you should choose a boot or shoe with a similar heel/sole design, i.e., neither too clunky nor flat, and the heel should be 1 inch to 1 1/2 inches high. Work boots and boots used in rainy weather may be appropriate, but boots with aggressive treads are not ideal as they can hinder your ability to get out of the saddle quickly. Not shown here are tall boots, which are also a great option for riding.
The Riding club does provide helmets for you to use until you get your own.
To protect against abrasion in case of a fall, you must wear long pants when you ride a Hoofer horse. However, the type of long pants that you wear is entirely up to you. Jeans are acceptable, but after a while, you may find that they are not the most comfortable option. Breeches and/or riding tights are designed specifically for riding. They don't have a seam that runs along the inside of the legs, and they are designed to stretch so they don't move while you ride. They will usually be reinforced by either a full seat or knee patches with an extra layer of fabric or a suede-type material.
If you still prefer jeans or other pants with a seam up the insides of the legs, but find that you're being rubbed raw by the stirrup leathers, chaps or half chaps are another option besides breeches. Still another option is lightweight sweat pants or leggings, if you don't want to purchase breeches; the elastic cuffs and stretchiness helps keep sweats and leggings in place over your ankles.
Chaps and/or half chaps are more about comfort than safety, although chaps can certainly protect you from abrasions more than jeans or breeches. (And chaps can keep your legs warmer in the winter.) Chaps, half chaps and tall boots perform many of the same functions, although tall boots are dressier and can be used for shows (and are more expensive and harder to fit).