Dying your hair doesn't cause it to break off (if you take care of it properly or if you don't follow the instructions). Bleach which is different then dying your hair, bleach can cause more damage then hair dye if you don't know what you're doing. If you're not confident with dying your hair at home, maybe try convincing your mom to take you to a salon to get semi permanent or temporary red hair color, and see how you like that.
If you don't take care of your hair properly, by using products that are meant for color treated hair then you can cause a bit of damage, but not to the point where it would break off. Maybe try a semi permanent dye to start out with, these usually wash out in a few weeks. But if it's darker or brighter then your natural color, it might stain your hair. I was about your age when I started dying my hair, I started out using semi permanent hair color. I'm now 42 , I bleach my hair but I haven't had it break off.
Just to warn you red fades very quickly, you would have to touch it up about 4-6 weeks or when you notice it's fading. Red can also be hard to remove, you have to bleach it out if you use permanent dye, or try to dye over it. Also dying your hair can get expensive, even if you don't go to a salon.