Agreed with everyone else- get your own helmet and boots! Proper head and footwear are critical for your safety, so do it as soon as you can. I don't know how serious you are about riding, but if you're not 100% sure that you're going to stick with it forever (or if you're growing, idk how old you are) then I'd suggest buying a cheaper pair to start with.
It's good for barns to keep helmets around for new students that don't want to commit to buying a helmet for a few lessons. However, you don't know who's been wearing these helmets, how old they are, and if someone's had a fall in them. You should always replace a helmet after a fall in which the helmet hits the ground, and they should be replaced at least every 5 years. Most hiking boots have thick soles and very textured tread and not necessarily a heel. They're designed to have very good traction. By this design, they aren't great for riding. If your boots don't have a heel then your foot could get caught in the stirrup in case something goes wrong. If you have a fall then the tread could prevent your foot from leaving the stirrup and you could get hung up. You've even said that you can't feel the stirrups properly in these boots, so it's time to buy something more proper for riding.
Here are a few options to get you started with your search...
This is the helmet I have. I've been riding for years and could afford to buy a more expensive helmet if I wanted, but I'm very satisfied with this one. It's attractive, low profile, inexpensive, and comfortable. Ovation® Deluxe Schooler Helmet | Dover Saddlery
I have one of these that I keep around in case I have a friend riding with me. The quality is definitely less, but I'd rather see someone wear something like this than nothing or a sketchy helmet: Troxel® Sport Helmet** | Dover Saddlery
These are the boots I have, and I love them! They'll last you at least a few years of weekly lessons, though there are less expensive options out there. Ariats are popular, and I'd recommend them: Ladies' Ariat® Heritage III Zip Paddock Boot | Dover Saddlery
I also have these that I keep around as a spare. They're easy to care for and cheap, but definitely lower quality. I don't use them that frequently, but I'm not sure that they'd last forever with frequent use: Saxon Equileather™ Zip Paddock Boots | Dover Saddlery
Have fun with your search! For under $100 you can get some basic equipment that will make your riding safer and more enjoyable. For under $200 you can get the same equipment that I use myself, and it'll last you a long time doing regular lessons.