My apologies. Got caught up in some things. I've written down all the suggestions I've been given so far in a notepad. I guess I have to say that I do enjoy slice of life stuff. At least I did with film (really enjoyed Whisper of the Heart, Girl Who Leapt Through Time stuff like that.).
So I'm totally fine with that. I don't mind action stuff either. I guess I should probably check out Deathnote at some point. Sorry if that was a vague response. I guess I'm not too stubborn in my ways. I know I've been critical about Attack on Titan, but it's more so for the pacing, and how draining it is when all the aspects that I find flawed come together. But I'm not against action/serious anime. I guess I would just hope that it has good writing/good pacing (all you can really ask for I guess).
If you're basically open for anything then I'm going to list what I recently suggested for friends, so if you've seen some of the stuff in here simply ignore it...
Action/serious:
Hunter x Hunter- shonen action. Very DBZ styled
Fate/Zero- Seinen (aka made for older audiences). Has an episode or two where it's largely talking but it's not the norm.
Berserk Golden Age Arc Trilogy- film trilogy. Amazingly good, though the experiment with the animation didn't pay off truly until the second half of the second film. Seinen and similar to Attack on Titan of dark fantasy setting.
Cowboy Bebop- you were asking for great classics and nothing beats Cowboy Bebop in that regard. Another one of my favorite anime series ever. Probably my favorite.
Fullmetal Alchemist Brotherhood- Action series similar to Attack on Titan in the sense of not being styled as other shows with story arcs. But it has good pacing unlike AoT.
Samurai Champloo- from the creator of Cowboy Bebop. Also amazing, some people prefer it to Bebop. Combines a bit of hip hop and modern culture with samurai setting.
Psycho-Pass: Dystopian sci-fi setting. Very well drawn, arcs are interesting and while the main cast left a bit to be desired they were still very thorough with the two leads.
Ergo Proxy- Very good sci fi and fantasy series. Also a bit dystopian. A bit more artsy in the presentation than some of the others I've recommended here.
Death Note- I see you haven't seen it. Give it a watch a very amazing anime.
Monster- An anime that outside of being drawn, doesn't have much to do with the medium. Amazing thriller and getting an HBO remake at some point in the next few years.
Baccano!- a darling of many people here. Definitely recommended here. Used to be in Netflix not sure if it's still there.
Wolf's Rain- a flawed yet very emotional anime. I believe this was the start that made me cry very easily with shows.
Gintama- the bigger of my recommendations as it has 200+ episodes but definitely a good comedy with a sprinkle of serious arcs. If you want something that's way ahead of you with no fear of catching up, watch this.
Black Lagoon- amazing seinen action series. Definitely a must watch.
Trigun- a classic in the action series.
Gun x Sword- while not a classic per se, it's still an oldie and a goodie. I'll probably rewatch it soon enough.
Katanagatari- amazing series. One of the many you must watch.
Slice of Life/Comedy:
Daily Life of High School Boys- one of the better slice of life comedies out there. A series of skits but they're all amazing.
Nichijou- Basically a random collection of skits all revolving around a high school, specifically a group of high school girls. If you want a smile on your face this'll help out immensely.
Kids on the Slope- from the director of Cowboy Bebop comes this slice of life about high school kids and jazz. Very charming and has a bit of drama. But what high school life doesn't?
Clannad and Clannad After Story- one of the more emotional slice of life shows there are out there. A hint of fantasy but not too much to take it out of this category. Watch this when you want something that can get serious and a bit depressing at times.
Great Teacher Onizuka- you mentioned classics, here's another. Amazing show.
Usagi Drop- a sweet slice of life story about a man who isn't married taking care of an unwanted child. Stop with the anime avoid manga AT ALL COSTS.
Nodame Cantabile- decently long musical show about people trying to go forward with their careers in the classical music business. Amazing too, it will not disappoint.
Ef a tale of Memories/Melodies- there are two seasons. Can't remember which one is the first name-wise, but watch them in order. Very good dramas/romance.
Kokoro Connect- make sure you watch the whole thing, including the OVA's that 'finish' the story. There's something like 17 episodes total.
Spice and Wolf- while not slice of life in high school per se, it's a very relaxing fantasy show with a very intricate love story.
Eden of the East- watch the tv series and the 2/3 movies that follow. Very good series.
Hyouka- an AnimeGAF favorite and I can't disagree. Kyoto Animation at its finest.
Detroit Metal City- hilarious anime with relatively short episodes about a metal band.
I have more shows if you see this and want more recommendations once you're done.