The first one you just have to see is Honey and Clover! It's about students of an art academy, about their lifes and the difficulties they face..about love and unreturned love ans also about a soul searching trip to find yourself again!
Another great one is Elfenlied...though it's very violent. It's about the creation of a superweapon that looks like a girl. the opening tune is already fantastic!!
At the moment I am watching Full Metal Alchemist (FMA). I've only seen 8 episodes but I understand why it's called the best ever made. You can't play for God, there will be consequences if you do. More i won't tell...it would spoil the pleasure of watching it.
Maybe the very best of all is haibane remnei. This serie deals with girls who have wings wings and a halo. They seem be be born from some cocoon and live together. Where do they come from, why can't they just leav. You have to discover the questions yourself. A beautiful anime with backgrounds that look like oil paintings. A serie that starts a bit childish but becomes a complex story of sorrow.