Melancholy of Haruhi Suzumiya. Earlier I watched the mainstream anime for kids but I wouldn't call those real anime and they didn't get me into it as much.
Well, the first anime I watched was Sailor Moon, and I loved it... but I'm not sure if that necessarily "got" me into anime. I think what did was Excel Saga. I remember renting the hell out of a DVD volume of it from the local library before I had the internet. Got me super intrigued in other anime.
Not counting anime I saw as a child, because back then I didn't even know the differences between Western and Japanese cartoons. But many years later, I was bored one day and stumbled upon this series called Love Hina on the internet. After I finished it, my soul was sold.