There are quite a few anime series you might want to try out, but I'll just give two examples:

Try Wolf's Rain if you're interested in a tragic narrative set in a post-apocalyptic world with some fantasy mixed in. The story is quite complex and full of symbolism, and the characters get a lot of character development and are quite memorable. The music is absolutely gorgeous by mixing beauty into tragedy with some ethnic influences. Animation looks quite good too, and many scenes are breahtakingly beautiful. It's well worth watching.

I suggest you check out Monster if you want to watch urban psychological horror with detective elements and great storytelling. It has a great cast of wonderful characters, the story is top notch, the animation is quite good, and music supports the narrative well. It starts out slowly but develops during its lengthy running time, and as you get through more and more episodes, you begin to see how seemingly random things are actually connected to a greater whole. The story plays with your mind and offers a lot of thrills while it also has mysterious elements which keep you on your toes.