The first anime that made me acknowledge anime on a large scale and start looking for more anime was probably Shadow Skill, which is a three part OVA. Probably pretty unusual as someone's first anime, so I'll go with Dragon Ball Z.
I would say Dragonball Z ... i never really knew what anime was, i just thought sailor moon and DBZ were just cartoons, then i stopped watching, then for some reason i started watching DBZ like religiously and got hooked from, there
Most of the anime such Tenchi Muyo, Sailor Moon, DBZ, Outlaw Star, Gundam Wing, that was aired on toonami got me into wanting to rent anime and then go on to buying them and having various titles on birthday and Christmas lists.
I love this question, brings back the good memories and the many animes I saw. The one that got me addicted majorly was none other then Tenchi Muyo. At first, I was afraid to watch this anime in the house due to the fact that my brother would think I'm a faggot. But that changed as he was watching it too and many other animes. So I was happy and I openly watched Tenchi without a care. I still love the anime and I got the most or majority of the stuff that is out for this anime.
Abenobashi got me into buying anime. And while Tenchi Muyo definately wasn't my first anime, it was the the one the made me decide to like anime, and not just cartoons (I had seen tons of anime that I loved before that though, Tenshi was probably just where I finally made the full distinction).
I remember coming home from school and watching Tenchi Muyo and Gundam Wing on TV. Those are the two shows that got me hooked. And then Blue Sub No.6 came on for like a week lol. That was also awesome. I of course have since learned not to watch anime on TV even though that's how I got into it.