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.
During the first ten years of this HOWTO's life, I reported that from a new user's point of view, all Linux distributions are almost equivalent. But in 2006-2007, an actual best choice emerged: Ubuntu. While other distros have their own areas of strength, Ubuntu is far and away the most accessible to Linux newbies.
You can find BSD Unix help and resources at www.bsd.org.
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.
I have written a primer on the basics of Unix and the Internet.
I used to recommend against installing either Linux or BSD as a solo project if you're a newbie. Nowadays the installers have gotten good enough that doing it entirely on your own is possible, even for a newbie. Nevertheless, I still recommend making contact with your local Linux user's group and asking for help. It can't hurt, and may smooth the process.