>>122374Full Metal Alchemist: Brotherhood I feel is a pretty decent way to get into anime as a whole. It's 64 episodes long, so be prepared for the long haul. It's completely worth it though.
The Ghost in the Shell franchise would probably interest you. It's got cyber-punk and it takes some episodes to expand on the world a bit. I recommend to either start with the first Ghost in the Shell movie (1995) and then watch Innocence [or] start with Stand Alone Complex season 1 and 2 and the Solid State Society and then watch whatever you didn't watch. And then you can watch Arise if you want. Most people don't like Arise, but I thought it was decent.
Steamboy isn't cyber-punk, but steam-punk is close enough. It's not perfect, but it does a lot of interesting things as well as phenomenal animation.
If you want fantasy, most Ghibli films are a good place to start. Spirited Away, Princess Mononoke, Castle in the Sky, The Cat Returns, Nausicaa, etc.
Death Note isn't exactly horror, but it's got a dark kind of art style and sense to it as well as interesting characters and premise.
I haven't watched Planetes, but it's probably as close to "slice of life in space" as I can think of without really looking around. Cowboy Bebop could also be described like that. Both the series and movie are fantastic.
For urban storytelling, Tokyo Godfathers fits the bill decently. It's a movie, so it doesn't get to explore a lot of the city, but it's a really great watch, one of my favorites. I think Patlabor might fit this "genre" as well. Start with the first 7 OVAs and then watch the two movies if you want. Haven't watched the TV show, but I hear it's good. Yozakura Quartet might fit this, too. Dennou Coil's kind of iffy, but I'll add it in as well.
Probably a little too overwhelming and wordy, but just know that all of those are good stuff.
