Welcome, and I just want to start by saying that you aren't the only late bloomer - I'm in the exact same boat as you! I've only been riding since this spring and I'm 18 :P
While I agree that your boots won't really affect your balance, I used to have cheap "fashion" leather boots that were literally smooth on the bottom and I had problems with my feet slipping in the stirrups when I would try to post. Proper paddock boots helped with that a lot.
It's normal to feel like you have no idea what you're doing at first, but you'll get used to the horse's movements the more you ride him. As others have said, make sure the ball of your foot is on the stirrup, and keep your heels down. One thing that I found immensely for me was when someone explained that you shouldn't keep your heels down with your ankles but rather let your weight sink into them. It'll help act as a shock absorber when the horse moves, so keep your ankles relaxed.
Your instructor will give you tons of tips as you go, but in general just relax your body, keep your back straight and your shoulders and chest open, ankles relaxed and make sure that you BREATHE. I still have trouble with that one, but it'll keep your core relaxed and flexible, which allows your seat to move freely of your upper body, and it'll also make you and your horse more relaxed and at ease.
Welcome to riding, and I'm glad I'm not the only hopeful jumper that's just started