self defense is more about having a strong sense of situational awareness, level headedness and attention to detail. its an excellent question you ask. 13 is a great age to being excercising your body.
Karate means the way of the empty hand, japanese 'hard' martial art. Emphasis is learning controlled breathing techniques and physical routines practised ritually with an observance for tradition that values respect and discipline. The physical routines will train mostly in striking and blocking so that these moves become a second nature to implement at the speed of reaction if you are ever in a closed quarters unarmed brawl with someone and no immediate means of escape.
There are many schools of karate, some teach using archaic weapon types to help develope hand, eye coordination and 'muscle memory'. Among the different schools there will be found different specialist aspects that the practicioners train for. I would be very wary of any school of karate you might consider applying for. Do research on the lingage of any study and understand at least a basic grasping of cirriculum of what you will study and train to do at low, medium and advanced belt levels...if the school is indeed sincere in its persuit of an ideal technique of mastery then they should have no trouble explaining that to you. Do be very wary of dojos that want to offer a little of something for everybody, as often that is what you will end up training and mastering if anything at all.
It has been estimated that 1 out of 3 people in thier life time on average will be in such an altercation where application of prompt empty handed technique can prevent a lot of trouble or trauma or even save a life.
Tae Kwon Do is a 'hard' korean martial art that was an asserted assimilation of many local schools to the Korean peninsula. TKD is very streamlined and tracable legally. It can be a much simpler art that a traditional karate practice because of the transparency of the cirriculum which no respectable school as such will hesitate to fully inform you of. Mostly, from what i have witnessed, ranking and belting up will mostly consist of proper performace of certain 'forms' or routines to certain tolerances of standard. These form routines (like a coreographed dance almost) will demonstrate your balance, speed, power and agility of execution. TKD is probably most known for its very dynamic and pragmatic kicking routines and is one of teh few arts out there that is avidly engaged in the world olympics as a sport. If you train TKD you will train hard. you will stretch hard, you will spar alot, you will learn balance, preccision and technically breaking down exactly what you are doing. You will not move beyond the 'straight punch' until you know exactly what you are doing with a straight punch for example, and can at least intellectually grasp exactlly the vision of a standard that you need to train to individually to make it a truly effective technique. In TKD expect to go running or encouraged to do so to improve your cardiovascular endurance and power.
Overall, between the two i would reccommend TKD.
Karate is a deeply bastardized and strung out tradition overall. When it comes to how it is marketed in the west, i feel that finding places one might really want to study it are few and far between to get thr authentic flair of one complete tradition. Reality is you will find many places 'Mc Dojos' We call them because they are businesses, they offer compromised routines and patch work broken tradition and legacy, there is a common sight you will see small children that probably couldnt fight thier way out of a wet paper bag cycling through more and more belt tests to the next color rank, this is admissable as a 'self esteem' and discipline thing for young children who are in the position to look up to thier elders, of which thier intructors are included. Truth is, its just a business, and what you put in, for yourself, is what you will get out of it, for yourself.
So, at least go for something that is very clear and obvious the work before you to be done. Try TKD for overall great fitness. If you are hell bent on self defense, i reccommend combative jiu jitsu.