If you don't mind starting from the command line, OpenBSD can be a good and minimalist place to begin. [For instance, when you aren't familiar with the concepts of "package management," diving right into a system that presents you with a package manager is no fun.]
Of course, OpenBSD is also not very "convenient" until you know what you're doing, and these days Ubuntu is very much plug'n'play on most systems. So if you have the equivalent of an old 486 to throw OpenBSD on (really, anything from a 486 on up is more than fine) and a recent machine to poke at Ubuntu, you stand a decent chance of getting quickly up to speed -- and will also be exposed to common differences between "UNIX-like" systems (and BSD vs. the GNU world).
I'm not sure exactly what to suggest as reading material. I'll see if I can dig you up the title of a very straightforward "UNIX" book I have on the shelf, which is nice because it doesn't pretend to be something it's not (and doesn't pretend UNIX is anything it's not). I also seem to remember "Ubuntu For Non-Geeks" being pretty good if you're approaching Linux; it's more technical than the title suggests, but attacks the "things you'd expect from a desktop" rather than the "here's how to start Apache" problem-space. Immersion is useful because once you start trying to accomplish something, you'll actually remember it over time.
The thing is that most daemons and server tasks are pretty straightforward, but people will always make careers out of making software seem big, complex, and important - and some books are authored from that perspective.* Remember that UNIX became the lingua franca because, at heart, it's quite simple and has been easy to keep reimplementing.
*O'Reilly anything is popular and usually splits the difference... they're actually quite variable in quality depending on the author. For example, the first half of their TCP/IP tome was a good introduction ten years ago, but then spent the rest of its pages on BIND, sendmail, and SATAN (anyone remember SATAN?) specifics. It'll take a while to figure out what's a fad because-it's-a-fad vs. what's a fad because it's useful.
