I've seen people who are 10 years old or younger start martial arts, any age is fine. 13 would be a fine age for you to start, considering what you want to do.
From my experience, I would start out with your basic Karate (jutistu). This teaches you mostly about defense, while also exposing you to some fighting techniques. You may or may not want to try for a higher level belt, or maybe just get a basic belt in several different styles. After that, you can go in to Kung Fu, taekwondo, etc...
