There are several reasons I've heard for wanting to learn the language, and pretty much all of them were represented in the J103 class I took a year or so back. The people who have to learn it for work will do so whether they actually enjoy the language or not. The best advice for them really is classroom learning, due to the structured environment, and the fact that the teacher isn't likely to teach outside of the niceties.
For everyone that does not apply to, however, I can try to offer a little advice.
First and foremost, if you don't have a reason better than "well, I just kinda want to", I wouldn't really suggest trying. Learning any language, not just Japanese, isn't something that you can put on a hobby level akin to building models. The reasoning for this seems simple enough - that being, doing something outside of necessity requires a good helping of self-motivation... and unlike watching that model boat take shape (a tangible motivation), learning a language "just because" does not offer such immediate rewards. There were many people like this in J103, and not a single one of them went on to J104. (The caveat to this, of course, would be the person for whom learning other languages is a hobby.)
The interest has to come first. Sure, it sounds simple, and I suppose it is, but it's still a true enough statement for all that. My interest (unsurprisingly, amongst the kid through twenty-something age group) started with anime and manga, so I got into the language with that in mind.
Here's how I got as far as I have so far, in order, to the best of my recollection. Note that I don't really suggest going about learning the language this way, unless it's truly incidental to the enjoyment of entertainment of Japanese origin. You could call the following the hard way, or the scenic route, depending on your preference.
Some time in 1996, two of my friends independently suggested a couple of anime to me. At the time, I had no idea what made it different (we all watched English dubs in those days, since dub/sub did not come on the same VHS... yes, VHS tape, and the Japanese with subtitles version cost an additional ten bucks a pop), other than that it was generally much less episodic than what was normally shown in the various afternoon cartoon blocs. This appealed greatly, since it struck me as more akin to novels, as opposed to the short story structure of the cartoons of the day.
Two years later, give or take a few months, I started my first attempt at translating a manga. It almost goes without saying that I made negligible headway on that project. Not that it was a bad idea, really, so much as that I made a bad choice as to which manga to try to translate.
From there, anime on video evolved to the new media, DVDs, and at that time, the English dub track and the Japanese track were included on the same product for the first time. I'd like to be able to say that we made the choice due to some high-minded idea like wanting to watch the shows in their original form, but it's far more likely that we were just sick of hearing the same English voices miscast time after time, and switched to the Japanese language track for a change.
The rest went rather as one might expect, given the circumstances. Picking up a word here, a word there, and finally building something that might be called a vocabulary... though, certainly not one to be used in polite company. A lot of the trick in J103 was remembering which words I knew that I shouldn't use. Other than that, though, I was probably over-prepared for an entry-level course.
So, how would I suggest going about this, having my own piecemeal experience to look back on?
1) Starting with your interest is fine, but once you know it's not a passing fad for you, start taking classes. If you're not sure where to look for such a thing, your local community college is probably the best place to start. If you have a choice in the matter, I'd also suggest getting a native speaker as your teacher.
2) After that, supplement your education with whatever sources of the language you can find. Japanese restaurants aren't a bad place to try out your education and vocabulary, either. Old subtitled samurai movies aren't bad, either, though you'll get a lot of archaic word usage in them.
3) For the written language, once you have a base of the hiragana and katakana systems, I'd suggest trying to translate manga. Aside from the amusing stories, manga for younger people also tend to include furigana, which are small hiragana (individual syllables - ひらがな, for instance, which are the four hiragana that make up the word "hiragana") to the right of the kanji (advanced ideograms - 漢字, for instance, which make up the word "kanji") which give the sound reading. This makes the kanji in question easier to look up in some instances, since you can search for it via the given hiragana reading, instead of trying to pick out parts of the kanji itself and looking the word up that way.
4) In addition to the above, try to find a good set of kanji flash cards. Working with these during commercial breaks of your favorite show or sporting event seems to work pretty well, in my experience. At the very least, you notice the commercials less.