Grumble Grumble Grumble
I'd second Haibane Renmai as a good serious anime series. And if your into the Cyberpunk genre ('Akira'), the series 'Key the Metal Idol' (which makes a serious statement on the Japanese Idol singer phenomena) and 'Serial Experment Lain' are good choices. 'Boogie Pop Phantom' may also be a Cyberpunk style series you may like.
If your into religious symbolism, then 'Neon Genesis Evangelion' is an obvious choice. 'RahXephon' is also a series that is similar to EVA (and IMHO, is the better series...)
The works of the director Satoshi Kon are good serious anime works, such as the movies 'Perfect Blue', 'Millennium Actress', and 'Tokyo Godfathers' (The last two were runners up to the Best Animation Feature Oscar that 'Finding Nemo' took..) and the TV series 'Paranoia Agent'
Other Anime movies that are good choices are, there's 'Jin-Roh the Wolf Brigade', 'Vampire Hunter D: Bloodlust', 'Blood, the Last Vampire' and 'Metropolis'