I highly, HIGHLY recommend starting with Avatar: The Last Airbender.
It is not anime: actually, it's an American animated show, a kid's show in Nickelodeon that ran for three seasons before being brought to a close by choice of the creators: it's an epic, emotional, and truly amazing fantasy show that is easily the best I've ever seen.
Now I know it's not anime, but it serves as a good bridge. Often, it's just hard to get used to such maturity within an animated medium. Avatar is a kid's show, but though it starts out expectedly juvenile, it matures over time and becomes a great show. I watched it before I started watching anime, and it helped me be a bit used to it before I started on the anime itself. It's a great bridge to cross before starting in on it.
Then I recommend Death Note (psychological thriller), Full Metal Alchemist (fantasy/action/drama), and Clannad (comedy/romance/drama). These three are, in my opinion, the pinnacles of their respective genres, and will give you a good idea of what to expect from anime of different genres. Also, Clannad will make you cry (specifically the second season, After Story).
Other miscellaneous recommendations are Cowboy Bebop, Orphen (the first season), Darker Than BLACK, Samurai Champloo, and Fooly Cooly (this is a weird one: I recommend getting used to anime in general before starting this one).
EDIT: Also, Trigun. Fantastic anime, even if it has one of the most disappointing endings in the history of endings. But the show is awesome, and more than worth it.
Also, if you've up for a Japanese kid's show, Digimon Tamers (the third season of the show) is actually really, really good. A bit kiddy at first, but sometimes gets rather mature, and tells a great story with awesome characters.
It is not anime: actually, it's an American animated show, a kid's show in Nickelodeon that ran for three seasons before being brought to a close by choice of the creators: it's an epic, emotional, and truly amazing fantasy show that is easily the best I've ever seen.
Now I know it's not anime, but it serves as a good bridge. Often, it's just hard to get used to such maturity within an animated medium. Avatar is a kid's show, but though it starts out expectedly juvenile, it matures over time and becomes a great show. I watched it before I started watching anime, and it helped me be a bit used to it before I started on the anime itself. It's a great bridge to cross before starting in on it.
Then I recommend Death Note (psychological thriller), Full Metal Alchemist (fantasy/action/drama), and Clannad (comedy/romance/drama). These three are, in my opinion, the pinnacles of their respective genres, and will give you a good idea of what to expect from anime of different genres. Also, Clannad will make you cry (specifically the second season, After Story).
Other miscellaneous recommendations are Cowboy Bebop, Orphen (the first season), Darker Than BLACK, Samurai Champloo, and Fooly Cooly (this is a weird one: I recommend getting used to anime in general before starting this one).
EDIT: Also, Trigun. Fantastic anime, even if it has one of the most disappointing endings in the history of endings. But the show is awesome, and more than worth it.
Also, if you've up for a Japanese kid's show, Digimon Tamers (the third season of the show) is actually really, really good. A bit kiddy at first, but sometimes gets rather mature, and tells a great story with awesome characters.