First of all, make your goal "learn about computer security" rather than "learn how to break into other people's computers." Even if you are really dead-set on the second goal, you should attempt to achieve the first goal, because then you will develop a deep understanding of what you are doing instead of becoming a script kiddie.
Anyway, here's the advice I would give about how to learn about computer security. (This is pretty much how I went about doing it, except that some steps have been changed in order from the way I actually did it to the way I should have done it.)
First, learn C really well.
Learn to use Unix. (The lessons you learn are applicable to other operating systems too, but Unix provides a better environment for learning this stuff.) Reading the man pages of common commands will teach you a fair amount about how the system works. The man pages are kind of like Wikipedia, in that by reading one page you will find references to other things that you need to know. Get a book on the design of the Unix operating system (e.g. The Design of the Unix Operating System
), read it, and understand it.
Go through djb's "security holes" course
slides, and understand them. Where details are missing (because they're just the slides from a course, not a complete course), fill them in. (Note: I thought that the first half of the course was significantly more difficult than the second half! So you might want to skim until you get to something you can understand, and go back later.) There may be an equivalent course at your university.
Take a class in network architecture or "computer systems", if your university offers it. This will hopefully introduce you to network protocols and such things, which you expressed interest in. I found that the unix man pages had poor coverage of these topics, so I needed to supplement my knowledge from some other source.
At this point, you'll probably know enough to figure out what bits of knowledge you still need.