Waha, mine is a long, sad tale of one, like many of us, who first got into the fandom through the original Saban dub of DBZ.
But I'm lazy and don't feel like detailing it.

It was probably 2000 or 2001, though.
The short(est) of it is that one random day I was randomly at my grandfather's house and I randomly turned on the TV to randomly see a cartoon I had never seen before. ...Randomly.
This also, mind you, was back in the day before I really knew what 'Anime' was - I had been into the Pokemon fad when that first was a hit, but the only time I ever heard the word 'Anime' used was when I started watching Monster Rancher on and off on some-such Saturday Morning block. So it was just some random cartoon with one of those weird animation styles I had been seeing a lot lately.
The amazing thing is I happened to randomly catch one of the very first episodes - I don't know which it was, due to the way they conglomerated so much footage together and spliced episodes back in that dub, but I tuned in about halfway through the episode just as Radditz was kidnapping Gohan and
Gokou was teaming up with Piccolo. Obviously this is where Dragonball Z began, but I had no way of knowing that, and thus I figured since so much didn't make sense that I was missing a lot and picking up an episode in the middle of the series (ironically I later found out that I did not...and even later, that I actually
DID!!).
I found it interesting since typical cartoons were self-contained and I liked the continuing storyline I compared to as 'soap-opera like' (since back then I hadn't seen many series' that used the format besides the soap operas my mom watched) and the animation. So I started watching it and eventually got hooked. I was, as the typical 11-year old (I think about that age, it was 5th/6th grade I got into it) watching the series at the time, nonethewiser about the atrocities the Anime felt, and I wouldn't learn otherwise until the dub entered into that loop where all they did was replay through mid-Freeza Saga.
I got so into it I found fansites and everything, had a 'that's not
Gokou...' reaction when I started seeing images of him as a Super Saiyajin labeled as being
Gokou, then eventually got those really cheesy summaries to fill me in on everything that happened in the series past where I was stuck...noticed in said summaries that some character names and terminology was different...etc, etc, etc...and ended up finding more fansites to explain this shit to me.
So I happened upon the legendary DBZ Uncensored and I never went back.
Next thing I remember was in my 8th Grade Computer SWAP (School Wide Activity Program; something we did at the end of the day on Tuesdays/Thursdays that was basically like a mandatory school club, and I got into the computer one so we could just...use the computers for an hour at the end of every day) that I tracked down fansubs on another long-dead fansite and got hooked on that version even more. It left a lot of holes in my viewing experience, though, so to this day I've never seen about half of the mid-Cell Saga in Japanese, bits and pieces of the Buu Saga, the early half of the Freeza Saga (furthest back episodes the site had were Vegeta vs. Zarbon) and the entire Saiyajin Saga. Nor most of Dragonball since I never found fansubs of that when it was DBZ that everyone cared about. I've read all of the Manga, though (in Spanish ^^; ) but it'd be nonetheless nice to one day have the money/ambition with my dying DBZ fandom to be able to experience the Anime version of everything I ended up missing.
And that's my TL;DR story. ...So much for being lazy.

"I came to save you thanks to a magic bean. And if you think it was easy to find that bean, you're wrong."
--'Big Green' Yajirobe