Get one of the open-source Unixes and learn to use and run it.

I'll assume you have a personal computer or can get access to one. (Take a moment to appreciate how much that means. The hacker culture originally evolved back when computers were so expensive that individuals could not own them.) The single most important step any newbie can take toward acquiring hacker skills is to get a copy of Linux or one of the BSD-Unixes or OpenSolaris, install it on a personal machine, and run it.
Yes, there are other operating systems in the world besides Unix. But they're distributed in binary — you can't read the code, and you can't modify it. Trying to learn to hack on a Microsoft Windows machine or under any other closed-source system is like trying to learn to dance while wearing a body cast.
Under Mac OS X it's possible, but only part of the system is open source — you're likely to hit a lot of walls, and you have to be careful not to develop the bad habit of depending on Apple's proprietary code. If you concentrate on the Unix under the hood you can learn some useful things.
Unix is the operating system of the Internet. While you can learn to use the Internet without knowing Unix, you can't be an Internet hacker without understanding Unix. For this reason, the hacker culture today is pretty strongly Unix-centered. (This wasn't always true, and some old-time hackers still aren't happy about it, but the symbiosis between Unix and the Internet has become strong enough that even Microsoft's muscle doesn't seem able to seriously dent it.)
So, bring up a Unix — I like Linux myself but there are other ways (and yes, you can run both Linux and Microsoft Windows on the same machine). Learn it. Run it. Tinker with it. Talk to the Internet with it. Read the code. Modify the code. You'll get better programming tools (including C, LISP, Python, and Perl) than any Microsoft operating system can dream of hosting, you'll have fun, and you'll soak up more knowledge than you realize you're learning until you look back on it as a master hacker.
For more about learning Unix, see The Loginataka. You might also want to have a look at The Art Of Unix Programming.
To get your hands on a Linux, see the Linux Online! site; you can download from there or (better idea) find a local Linux user group to help you with installation. From a new user's point of view, all Linux distributions are pretty much equivalent.
A good way to dip your toes in the water is to boot up what Linux fans call a live CD, a distribution that runs entirely off a CD without having to modify your hard disk. This will be slow, because CDs are slow, but it's a way to get a look at the possibilities without having to do anything drastic.
You can find BSD Unix help and resources at www.bsd.org.
I have written a primer on the basics of Unix and the Internet.
(Note: I don't really recommend installing either Linux or BSD as a solo project if you're a newbie. For Linux, find a local Linux user's group and ask for help.)