Kids-10 to 12
Teen-13 to 17, I believe
Thought I'd just give a heads up about this, DD14 has tried out 2 times in the past (for the Teen tournament) and just barely missed it both times (you take an online test of 50 questions, and 35 makes you eligible to go on to the next level of auditions)- she was around 33/34. However, she is happy about this because both years she would have been the youngest person on the show and she is happy waiting until she is older to do it and actually have a great chance of winning.
I know this might not be great for all kids on here, but I thought I'd at least bring it up, if you have a kid that loves trivia and/or would like a chance to get to win some money (I believe there is a minimum amount of money you win, even if you come in last place).
Unfortunately, the next audition dates are still a few months away (usually in January or February) but if you have a child like mine who wants to study for it, that could be a good thing.

Does it cost a contestant to be on a gameshow?

I auditioned for the teen tournament when I was 15 or so. You had to call in to get an audition spot, then we drove 8 hours up to St. Louis for the actual auditions. I passed the paper-and-pencil test (which I assume has been replaced by the online test), but froze up in the next step, which is much more like being on the game show. (They ask you some interview questions, then you have to ring in and answer against 2 other auditioners.) It was fun - I had a good time doing it, even with the drive and with not being picked.

La Texican, you'd have to cover your own travel expenses to get to the filming, but there's no entry fee or anything like that.

I think it's actually 12. I read about it recently because a local kid was on the show. How fun!