Anime is a weird medium to get into. It's a style of animation from Japan that started to get mainstream in the mid 90's though Japanese companies had been producing anime for decades. Unlike western animation, which is largely viewed as a kid's medium, anime has several shows that are definitely not for kids with mature subjects or graphic material. Not to say that there isn't anime made just for kids but the medium gets it notoriety from the more adult series.
Probably the first anime series I ever saw was Voltron. However, the show that really caught my attention was surprisingly, Sailor Moon. It was a show mainly for girls but for some reason I couldn't help but watch it. It was probably because it was nothing like any of the shows I normally watched. Plus, it was aired in the morning before school so it was the show that got me started for the day. Soon after, Dragon Ball Z was aired and I focused my attention to that. This show was definitely targeted for boys so it was easier to justify watching it.
I've lost a lot of my interest for anime these days but there are a few that I keep track of. Inuyasha, FullMetal Alchemist and Ranma 1/2 are some of my favorites.