I received a mail from an acquaintance who'd come to the realization that his 13-year-old wanted to be programmer, specifically a games programmer. Here's the advice I gave. Perhaps others have things to add:
1. I'm tempted to tell you that the right way to learn to be a programmer is to start with LISP, or the lambda calculus, or even denotational semantics but you can come back to those after a few years getting your feet wet.
2. Lots of programming involves logic (or at least thinking logically) so learning about and enjoying logic is probably a good foundation. You could start by learning about boolean algebra since it's simple and fun and the basis for a lot of what computers do.
3. Since games programmer involves a lot of physics, you should also learn about Newton's Three Laws and Universal Gravitation and play around with things like springs and pendulums.
4. Basic trigonmetry is important to the games programmer. It'll be handy to know about Pythagoras and the relationship with sin, cos and tan.
5. Above all, start with a programming language and a good book and commence hacking: try stuff out, make little simple programs (even if it's a program that prints out "Hello" on the screen, or a program that prints out "Hello" ten times, or asks you for the number of times to print "Hello" and then does it). Just write code, whatever takes your fancy.
6. A good starting language is Python. Get the O'Reilly book Learning Python.
7. Python is dynamic so you'll be able to make progress very quickly, but for games programming you are probably going to need to get a little closer to the machine. And for that you should learn C by reading the classic The C Programming Language.
8. As you learn more there are some great books that will expand on what you can do: read Programming Pearls and The Practice of Programming. Think about getting: Algorithms in C. Read Structure and Interpretation of Computer Programs.
9. Also: avoid debuggers, learn to unit test. Debuggers are useful in limited circumstances, most code can be debugged by using your head and a few 'print's. Unit tests will save your life as you go forward.
10. When you are ready, try to write a version of the first ever computer game: Spacewar!
11. When your first company goes public think of me; I'll be an old man and probably won't have saved enough for retirement.