My first time was with Pokemon ^^ I really wanted to watch sailor moon as well but It was taboo at the time so I didn't.
Both my older brothers were into it and even more fanatical about catching them all than me, of course now they're both religious fanatics dedicating their lives to 'preaching the gospel' but there were a few good times back in the day, when I would let my imagination run wild. We even started to make our own Pokemon cards and rules... I actually felt like my brothers and I were getting along... but then religion of course fracked all that up.
Man I love Pokemon forever ^_^
I think I discovered Ranma 1/2 and Tenchi Muyo at about the same time. I was immediately hooked!
My first anime shows were Dragonball Z and Tenchi Muyo (and possibly Cowboy Bebop). More like ones that were shown on Toonami back in the Mid-late 90's. Back then, I still didn't get too interested in anime until InuYasha started to show on adult swim back in 2001-2002. Ever since then, I started to get hooked on it.
Sailor Moon, of course! Although I have vague fond memories of Card Captor Sakura and Pokemon, too...