Karate is Japanese, Tae kwon do is Korean. Karate is more hands and Tae kwon do is more feet, although both styles use both.
My son started Karate when he was 6 (because of the teenage mutant ninja turtles). It took him 7 years to get his black belt, but he is now 17 and helps teach a class at the local university.
Karate was wonderful for him. It's given him such self confidence. He's had a couple of occasions where punks at school have tried to pick fights with him and he just laughs and walks away because he has nothing to prove. As a parent that is perfect. Now part of me sometimes feels like telling him to make the punk bleed, but I don't.
A little advise. There are lots of schools out there. Most will give you some type of trial period for a set fee. Don't mean to be offensive to anyone, but there are schools that are "black belt mills". They test the kids for a higher rank every other month or so, sometimes every month and put out black belts in a few years. If that's what you want, great, but the whole purpose is self defense, confidence, respect and to be honest most kids I've seen at the schools take longer than that to develop those skills.
Be careful where the school makes you buy gear directly from them. You can get the same equipment at martial arts stores in your town or online from a couple of different martial arts/full contact sports websites. A good martial arts school is more interested in martial arts than the money. And there is no way a school should cost more once you become a black belt. In fact, most schools around here drop most, if not all fees, when people become black belts because they start to help out teaching etc. I would also ask around for opinions on the different schools. Someone you know or are aquainted with will probably have had some experience they can relate.
Hope this helps.