Here are some of the best typing games and tutorials and tests I've come across. I'd love your ideas and additions if you'll leave them as a comment.
Dance Mat Typing
This is a very accessible training program for first-exposure to typing. Pedagogically sound and fun for kids. The songs get old after a while and there's no way to skip them, but by then you can transition to games to increase speed--the foundation is laid.
These guys have some good, free games. I like them because I can instruct my kids to play at level three or higher and then they're off of the home row. They're typing full words rather than letter-by-letter, which is good because I don't think typing letters increases skills as much as typing words.
A rewarding shoot-em-up game. Not as pedagogically sound because you must press the Enter key between words (and there isn't variety built in), but kids with slower fingers can get into it until they get to 20wpm and be ready for Typer Shark. Kids like QWERTY Warriors, and I do too. There is also a QWERTY Warriors II that doesn't make you press Enter but it's not as adrenaline-producing. Try them both.
Once you're up to 20wpm, the best game out there is Typer Shark. Here is a LINK, but if it's blocked at your school you might try searching "Typer Shark" to find an open site. The great thing about this one is that there is variety built in, so it doesn't get old as fast.
Typing Master software provides this online typing test for free. It's the most accurate online test I've been able to find. We use the three minute version and track the kids' speed in a classroom wiki--kids put their speeds into a table on their own wiki page.
This isn't an online tool--just a suggestion. In our computer lab we used to use rubber keycovers, but they cost nearly as much as a cheap USB keyboard and are prone to tearing and loss. So now all our lab computers have two keyboards plugged in--one normal one and one where the 15 most common letters have been painted over with a paint-pen.