(T_T) ~sob~ I'm SO old...
What really got me into anime were three shows I watched as a child, and actually discovered pretty quickly that they were from Japan:
Marine Boy, Kimba the White Lion, and the ever-present Speed Racer. ...you have to keep in mind that I was watching Speed in maybe the first or second re-broadcast of Speed Racer(you do the math, I'm not sayin any dates), and loved both Speed and Marine Boy (Kimba was ok, but not as good)... There was a long dry stretch where you only saw what was on tv, then rarely a vhs tape, or imported book (Macross movie book, man it was gorgeous!), then finally started seeing anime at Sci-fi conventions, and finally the world became a good place, and the interwebs gave me my first full downloaded series: Bokusetsu Tenchi Dokuro-chan! (Thank you fansub.tv)... finally torrents... and HERE! (YAY!) I guess that about covers it... heh. (^_^)