[Warning, heavy use of wiki-links]
To start with, avoid ALL meta-series. They are not a good starting point, and are known for turning off people new to anime.
Meta-anime includes: Dragonball: Anything, Bleach, Naruto, One Peice, Sailor Moon, Ranma, or anything else that has a 3 digit episode list.
That aside, if you intend to start somewhere, good movies and short series are probably the better way to hold your interest.
Anything from Studio Ghibli is good. Link [http://en.wikipedia.org/wiki/Studio_Ghibli#Films_.28excluding_short_films_or_Ghibli_Museum_releases.29]
As are movies such as Ghost in the Shell [http://en.wikipedia.org/wiki/Ghost_in_the_Shell], Akira [http://en.wikipedia.org/wiki/Akira_%28film%29], Steamboy [http://en.wikipedia.org/wiki/Steamboy] and the Evangelion remakes. [http://en.wikipedia.org/wiki/Rebuild_of_Evangelion] (See: Neon Genesis Evangelion [http://en.wikipedia.org/wiki/Neon_Genesis_Evangelion])
Anime series are always going to fall down to a persons taste in both the art style and the content, so rather than going into detail on them, I will just make a few suggestions.
Martian Successor Nadesico [http://en.wikipedia.org/wiki/Martian_Successor_Nadesico]
Cowboy Bebop [http://en.wikipedia.org/wiki/Cowboy_Bebop]
Last Exile [http://en.wikipedia.org/wiki/Last_Exile]
Fruits Basket [http://en.wikipedia.org/wiki/Fruits_Basket]
Lucky Star [http://en.wikipedia.org/wiki/Lucky_Star_%28manga%29]
That should be varied enough without going too far.
[Edited for content]