What should you bring with you/wear to your lessons? One of the best ways to answer this question is: "Consult with your instructor." Most riders and trainers have very firm opinions on what constitutes safe and appropriate gear. And they can usually give you recommendations and advice on what and where to buy.
Now I am not talking about clothes to wear in a horse show right now. That is an entirely different matter. Show clothes are meant to enhance the elegance and beauty of your horse and your riding. Lesson clothes are meant to be 3 things: practical, comfortable, and safe. In the pursuit of order, I will start at the feet and work up.
This is a horse's foot. Also called a hoof. Each horse has 4 of them. They are large, and heavy. Sometimes horses don't watch where they are walking and step on things they shouldn't. Frequently these things are a human's feet. From this I recommend two things. First, pay attention to where your horse is stepping, and try not to be in the same place at the same time. And secondly, wear boots. Barefoot is unacceptable and so are open toed shoes and flip flops. A stable is not place to display your pedicure. Sneakers are OK for working around horses, but boots are better. (Sneakers are not appropriate while riding as they can cause your foot to get stuck in the stirrup.)
Current fashion amongst my students decries a black zip up boot. Check with your trainer for what they prefer. Paddock boots are the most comfortable and practical of the riding boot options available, and in the case of young children can also double as show boots. (If your child hasn't waded through the mud in them.)
Half chaps are a marvelous invention. Based in part on the full chaps that riders, both English and Western have worn for quite some time (still considered show attire amongst Western riders) and the design of the tall boots preferred by English and Dressage riders. It is the best of both worlds. The design allows the chap to be more flexible and comfortable than a tall boot, with more freedom than full chaps allow. (Definitely helpful when running to the rest room.) Half chaps come in many colors and varieties, leather and suede. The colored ones with cute designs look adorable on small children, but once you get past a certain age, the standard is that you pick a color that matches your paddock boots. I always recommend the zippered ones to the velcro, as the velcro catches hair and shavings quite quickly in a barn environment, and cease to fasten properly. Also, while some variants are thinner and therefore cheaper, the quality ones are preferred as the thicker suede/leather protects your legs from saddle rubs much more effectively.
As I said, some stables require a professional looking polo at all times, but that is not necessary. Any weather appropriate, well fitting top will work. Long baggy clothes that you sit on when posting are detrimental. And for anyone who likes the t-shirt above, it can be purchased here.
The picture above is a bad helmet. I say that unequivocally. The first point is that it will not double as a show helmet. Very few people have more than one helmet, so there is no point spending money on a helmet you cannot take to even an unrated schooling show without looking amateurish. Secondly the fit on these is very poor. The shape is not right, and I can tell you that I have seem them just pop right off a child's head (and choke them around the neck with the chin strap) when they were posting or cantering around the ring. Many of the Troxel helmets (say it now, don't buy Troxel) have the alluring feature of being "adjustable," this only makes matters worse. No matter how hard you try, the dial either gets spun too tightly and the helmet perches atop the rider's head, or it loosens as they ride and they end up with impaired vision as the helmet slips down over their face. I have seen more attractive adjustable helmets that are just as bad, so don't be fooled by a velvet cover either. The final problem with adjustable helmets is the fact that the dial by the back of your head is made of a hard plastic. If you should fall and hit your head, frequently the plastic breaks and you have a sharp jagged piece of plastic by your head. By all means be economical with any other aspect of your gear, but please, please protect your head.