I'll suggest a few that I've enjoyed.
Key The Metal Idol
After her grandfather's death, Key learns that she needs to make a lot of friends to become human. How many friends?
30,000
. Key leaves her small village for the city in hopes of becoming a well-known idol.
13 episodes and two 90 minute films (Drama with some action)
https://www.youtube.com/watch?v=0d1LdTJoZ98
Serial Experiments Lain
Lain, along with some classmates, receives a message from a girl who recently killed herself, saying that she is in the Wired, their version of the internet, and has met God. Lain is lead down a path that takes her into the Wired where she discovers many disturbing things, like the line between reality and the Wired are fading and... other versions of herself? Or are they? It's hard to give a basic summary of this series.
13 episodes (Cyberpunk, Drama, Psychological- and Techno-horrors)
https://www.youtube.com/watch?v=MM8RufZr5lw
Haibane Renmei
A young girl awakens from a cocoon after dreaming of falling from the sky, without being able to remember who she is. She is greeted by other girls who have wings and halos who have also come from cocoons, but with other dreams. They live in a giant city surrounded by a wall that's too tall to see over and no one leaves or enters, except for a small mysterious group. The young girl is given a halo and grows wings of her own.
13 episodes (Fantasy, Drama)
https://www.youtube.com/watch?v=ztSVbXekz0Y
Brigadoon
Marin, a young girl, is attacked by monsters who come from a mysterious world that suddenly appeared in the sky, and gets an alien swordsman protector named Melan Blue that appeared from an ampoule. It seems that the two have a connection, and find out why Marin is being attacked.
26 episodes (Comedy, action, sci-fi)
https://www.youtube.com/watch?v=_CszEg9oZl4
Without truth, there is nothing.