@LacrimosaRaven , get well soon!
Your teachers say that you should focus on your friend who's been hurt. I do agree that your friend should know the truth, that someone twisted your words and that you never badmouthed them. But something is bothering me here - why did nobody confront you directly? Why did they all fall for the old "that person said this about you" trick? If at least one person had come to you and said "come here, why did you say this about that person?", everything would have sorted itself out much more easily.
There might be a reason behind what they did though (or another truth you're not aware of). They just reacted with too much haste and didn't give you a chance to speak. Don't do the same mistake with them, try to be the bigger person.
Also, I know it sounds a bit weird, but don't seek revenge from the person who's been badmouthing you. Usually karma sorts out these things for you in a much better way that you'd have imagined yourself. Try to not harm/badmouth/etc anyone, but just foil the plans of the girl who badmouthed you. Did she want you to lose your friends? Well, you just said you made some new friends (Just be careful with them too now!). Did she want you to feel horrible? Pretend you don't care, at least when she is around - you only care about your friend who got hurt, not yourself.
Usually, people who try to harm you for no reason feel inferior and/or threatened by your presence. They most likely shouldn't (from our point of view), but their insecurities are usually too complicated for us to waste time with. Try not to fall into their traps. Try to do nothing wrong on your side, don't reduce yourself to their level, just stand up for who you really are. Soon they'll need to be held accountable for their actions, and that will be the mere consequence of their actions themselves. That's when we grab some popcorn and soda, sit down and relax, and call this "karma".
EDIT: When I was in school and felt frustrated about my classmates, an older friend of mine said "Don't worry about them. One day, you'll be a successful lady and they'll be serving you burgers at McDonald's". It wasn't McDonald's. But we made our life choices ever since we were in school, and those choices made a big difference not long after we graduated.