edit: feel free to reply no matter how old this thread gets!
I'm not particularly into anime, I like it just as much as live action shows or other types of animation. Having said that, I feel like watching some and I'm not sure what to pick.
Here's a list of other anime shows I've watched/manga and novels I've read, so you can better judge what to recommend. ( would greatly appreciate the story to be completed, I hate waiting years for closure) :
Full Metal Alchemist, Naruto : both great stories, I've read the mangas and loved them. They wouldn't be a priority to watch because I already read them.
Attack on Titan: started with the anime, it was great, switched to the manga, got extremely disappointed with the story development and its ending. It felt a bit like nonsense for me.
Dragon Ball/Z : probably my favourite, I've rewatched it recently. I don't like GT and Super.
Overlord: I've read good part of the novel and manga. I'm liking it so far
JoJo's Bizarre Adventure: please No thanks. Not for me.
Death Note: loved the first arc, not a fan of the ending but no hate for it unlike with AOT.
I also recall Escaflowne and Inuyasha, they were alright but not something I've loved or hated. Cowboy Bebop also in this category, excellent visually and great OST but the story wasn't as gripping.
Other than that... Feel free to suggest any other titles! Extra thanks if you are specific about any versions and why is the show so great.
My recs, roughly in order of what I'm predicting that you'll like the best, based on what you said.
Tengen Toppa Gurren Lagann - mechas, action, epicness, plot twists? Yeah, you got all of them, while you follow the story of a bunch of people raised underground, fighting their way up.
Chainsaw Man - fast-paced action in a world full of demons. The protagonist merges with one, after living in shit since childhood, and that changes his life. Expect lots of humour, and the intro becoming an earworm.
Kimetsu no Yaiba - historical-ish setting. The protagonist is a demon killer, who's sided by his demonised sister (yup). Sword fighting, great visuals, interesting and atypical story.
Dr. Stone - humanity has been petrified for thousands of years, and a single genius boy is trying to discover why, how, and how to revert it. The science there is 90% accurate, 10% fantasy.
Fullmetal Alchemist: Brotherhood (2009) - you watched the "original" FMA (2003), right? This one has slightly better worldbuilding and plot, and it's better grounded in the manga, although the homunculi are more relatable in the original.
Noragami - the story of a stray god, a girl who can see spirits and gods, and the spirit of a dead boy.
Mairimashita! Iruma-kun - protag gets sold to a demon by his parents. And the demon actually treats him better than they did. So he goes to school, alongside demons, while pretending to be one.
Dungeon ni Deai wo Motomeru no wa Machigatteiru Darou ka - a boy who wants to be a hero, fighting in a dungeon full of monsters, in a world where the gods descended and granted their blessing to mortals of their liking. Which gods? Yes. Romantic comedy with epic vibes.
Kill la Kill - watch this one if you watched TTGL and liked it. Heavy nudity, but the same "vibe" as TTGL. Hard to explain without giving you spoilers.
A few classics that I feel like they should be mentioned, but don't relate well to what you said that you liked:
Sen to Chihiro no Kamikakushi - movie, not a series. About a girl who's spirited away to a world full of yokai, trying to save her piggyfied parents.
Tonari no Totoro - also a movie, not a series, about children who move to a suburban neighbourhood full of magical creatures.
Neon Genesis Evangelion - psychological drama pretending to be a mecha anime.
Some old style isekais, since you mentioned that you liked Overlord:
Log Horizon - also "transported into a game", except that it happens with a fuckload of people at once. Most focus is how those people organise themselves in their former MMORPG, now real life, as well as how they handle the NPCs around them.
Isekai Maou to Shoukan Shoujo no Dorei Majutsu - two girls in a fantasy world decide to summon someone, hoping to enslave him to help them out... only to get their magic deflected and end enslaved. The world is the same as the game world that the protagonist (the summoned guy) played.
Re:Zero - protag gets transported to a fantasy world. He doesn't know how or why. His main ability is to go back in time, when he dies; so he experiences death over and over and over.
Mushoku Tensei - the protag starts as completely scum, but he gets slightly less worse over time. Great worldbuilding and characterisation.
Benriya Saitou-San, Isekai ni Iku - a handyman gets transported into a swords-and-magic world. Big focus on character relationship.
I don't recommend the anime, but check the manga if possible:
Elfen Lied - a discussion on what makes a human "human", morals, and the likes. The anime adaptation is rather poor to be honest, but the manga is decent. Content warning: gore, sexual violence, a girl pissing herself right off the bat.
Hoshi no Samidare - a great story, but poorly animated, about a bunch of people trying to save the world from a giant biscuit hammer that you only see if you know about it, that's going to hit Earth and destroy it. While subordinated to a girl who actually wants to destroy the world.
EDIT: GreenZanbato mentioned Tensei Shitara Slime Datta Ken, or "That Time that I Reincarnated as a Slime". It's also an isekai, roughly with the same vibe as Overlord; but the protagonist reincarnates as a slime, and he's far more self-aware than dummy Ainz. The novels are also great to read.
If you like DBZ you would probably like Hunter X Hunter and Yuyu Hakusho. These two are both written by the same person and have similar vibes but the story and setting are very different. HxH especially is very widely acclaimed for good reason. In this vein you may also like Black Clover.
Anime that I'd recommend in general would be Akame Ga Kill, Mahou Shouju Madoka Magicka (watch 3-4 episodes, trust the process), Parasyte, Gurren Laggann, One Punch Man, Angel Beats, Katanagatari, Made In Abyss, and Berserk (1997),
If you want to get in your feelings: Erased, Kotaro Lives Alone, and The Promised Neverland (season 1 only). Spy X Family (cute and silly but very good)
Some current very popular DBZ-esque anime is My Hero Academia, Jujutsu Kaisen, and Demon Slayer Kimetso no Yaiba.
Haikyu is great. It’s about volleyball and follows the same sort of theme of DBZ about constantly pushing to improve.
If you liked the environment and feel of Cowboy Bebop you might enjoy Samurai Champloo. Same director but replace jazz with hip hop and space with feudal Japan. I think it’s a masterpiece.
These two show might not be like the other shows that your have watched yet these two or definitely my favourite.
Odd Taxi - Generally anime cast has this typical behaviour where their talking or behaviour would not be realistic like irl (for example, Full Metal Alchemist: Brotherhood). You might not notice it now or might have noticed it but odd taxi is totally different from all the other anime that I have watched in regards to this. The plot is simple yet the way it moves is totally interesting. The characters are written well for a 12 episode anime. You will definitely like it.
Ranking of Kings - I have not completed it yet but this anime is so wholesome and kinda moving ngl.
Since you have watched death note, watch Code Geass too. When I started watching anime, Assassination Classroom and Violet Evergarden were one of the anime that totally bought me in. You may like these too.
It sounds like you tend to like shows with good story lines, and don't seem to mind older shows/animation as much. Simply based on their story telling, these are probably my 3 favorites.
Monster (2004): The story revolves around Kenzo Tenma, a Japanese surgeon living in Düsseldorf, Germany whose life enters turmoil after getting himself involved with Johan Liebert, one of his former patients, who is revealed to be a dangerous serial killer.
Ergo Proxy (2006): Set in a post-apocalyptic future where humans and AutoReiv androids coexist peacefully until a virus gives the androids self-awareness, causing them to commit a series of murders. Inspector Re-L Mayer is assigned to investigate, discovering a more complicated plot behind it that involves a humanoid species known as "Proxy" who are the subject of secret government experiments.
Fruits Basket (1998, but the 2019 remake is miles better and more complete): This is a beautiful story about trauma, love, guilt, and belonging. Don't be put off by the typical highschool slice of life trope. This is probably the most wholesome, heartwarming show I have ever watched. I've seen it 3 times, and on the most recent rewatch I cried the most. Probably every 1 of 4 episodes. I kept realizing subtle plot details I had never noticed before. It has great rewatch value.
I recommend Mob Psycho 100. The anime fully adapted the source material and its made by the same guy as One Punch Man. Don't let the strange art style stop you. The animation team goes full budget in action scenes. It's a character driven story with a cast of charming characters.
Assassination Classroom - Just finished watching this series and it’s one of the most heartwarming meditations on teaching and education wrapped in a batshit crazy premise of junior high kids being forced to kill their teacher before he destroys the world at the end of their school year. Made me cry a few times.
I'll give you two suggestions at opposite ends of the spectrum:
Neon Genesis Evangelion is a wonderful show. It starts out as a giant robot series, but then moves in on examining how the characters are broken mentally (as in, they're already broken, not let's break stable people). Does have a tendency to depress people, but they generally feel it was a good experience.
Just about any Studio Ghibli movie. Nausicaa of the Valley of the Wind, Princess Mononoke, Castle in the Sky, My Neighbor Totoro, Kiki's Delivery Service, Howl's Moving Castle. Some of these are more kids movies, but adults love them. Others have more adult themes. Stick to the ones directed by Hayao Miazaki. Warning: Grave of the Fireflys will make you cry and leave you devastated.
If you liked death note you should watch code geass. It starts like yet another mecha anime but get just crazy on the way, one of the best surprise I had.
If you like mecha both Gundam and Macross are worth watching. In general I prefer Macross (Frontier 😍😍) , but there is pretty good Gundam series (SEED would be my favourite). Moreover, both are a set of relatively in dependant series so you don't have like 200 episodes to watch but get away with like 2 times a 24 episode serie
I'm gonna take some leaps
If you like the mind games parts of death note (since you seem to like the first part)
I recommend trying Kaiji. My personal fav and one that got me changing my expectation of anime.
If you like DBZ I assume it's the fighting. And even though the first assumption is people who like Anime probably will hate sports , sports anime are usually just tournaments for battling. If you want superpowers, I might recommend Kuruko No Basuke. I you want Mind Games, maybe One Outs. If you want something more modern, Haikyuu is a good shout. Classic sports anime would be Hajime no Ippo
A general shounen? Fate/Zero is pretty safe to try.
When it comes to recommendations, realistically there are lots of great anime movies to give yourself more context for what to watch
Ghost in the Shell
Akira
Castle of Cagliostro
Kara no Kyoukai
Nausicaa Valley of the Wind
These are all just based on what you've mentioned but they're still a lot of spectrum for anime you could enjoy but haven't tried
Legend of the Galactic Heroes is a series with many episodes in which "...two interstellar states...are embroiled in a never-ending war...focuses on the exploits of rivals Reinhard von Lohengramm and Yang Wen-li, as they rise to power and fame in the Galactic Empire and the Free Planets Alliance..."
This is one of the newer shows made by Shinichiro Watanabe (Cowboy Bebop, Samurai Champloo). Follows the crew of a small terrorist organization as they implement their various plots - lots of intrigue that makes you want to keep watching.
Megalobox
Follows a boxer's journey in a semi-dystopian future. Very mature and themes of adulthood, especially season 2. And I mean real life mature, not gore and titties. I've never watched an anime that I found so relatable as an adult human being. Music is also extremely top tier and I'm personally a huge fan of it's animation style.
Seems to be a confluence of One Piece popularity right now. If you start watching the anime you be caught up around the time the final airs - which is in 3-4 years, which goes to show how much you have to watch through.
Fairy Tail has kinda a DBZ vibe, but the fights are over with in one or two episodes, not 45. (Funimation)
If you like suspense and psychological thriller, and don't mind subtitles, Kaiji is an excellent watch. It's about a man who cosigned a loan with a friend, from a bunch of loan sharks, and has to play increasingly deadly games to try to regain his life. It will take you for a ride. (Crunchyroll)
I really enjoyed Dr Stone. An apocalypse occurs, petrifying the entire world. Humanity starts to slowly awaken, and they try to rebuild the world while fighting against an opposing group trying to start a new world order. At least for season one. I haven't cought up recently cuz my depression has been kicking my ass. (Funimation)
That's all I got off the top of my head, and I hope it helps.
Edit: If you like Kaiji, it has two separate side stories.
Akagi is set in the 70s, I think, and is about a man who is apparently a savant at Mahjong, and ends up playing against a mob boss. wagering his blood against the bosses fortune. The show is really good at trying to explain the rules to the uninitiated without being condescending, and still keeping it suspenseful and engaging.
The other is Mr. Tonegawa's Middle Management Blues, more of a slice of life workplace comedy, centering around one of the antagonists of Kaiji, and trying to manage his employees while pleasing his boss. I didn't finish this one, cuz I got the point about halfway through, but I had fun while it lasted.
And I just remembered Cells At Work, also subbed, but an excellent romp which I would call "edutainment," but don't let that dissuade you! It anthropomorphizes different cells of the human body, describing their functions in kinda a slice-of-life meets monster of the week format. The white blood cells are the knife-weilding defence force of the human body, cutting down Power Ranger monsters that never were. Again, don't get put off by the edutainment label, cuz the edu and the tainment both hit hard.
I'm a little disappointed that Hajime No Ippo wasn't mentioned. It's from the early 2000s, plays in the late 80s and it's about boxing. The writing and how fleshed out the characters are is unexpectedly good for that kind of setup and it's one of my absolute favorites.
Two great comical animes are golden boy and great teacher onizuka but they have a lot of cliche anime perverted jokes I can see putting off a lot of people especially when they're not deep into anime.
Then there's goblin slayer but you might want to look up the controversy about the first episode before you go blindly into it.
I'm also not a huge anime fan, so my interests in the genre are pretty selective. The anime shows that I tend to enjoy are ones that usually aren't ongoing stories, and wrap up their narrative lines within a season or two. That said, I absolutely loved Hellsing and Hellsing: Ultimate (which is a sort of "reimagining" of the original series, if I recall correctly). They've got some of the most entertaining action scenes I've seen. The CastleVania series on Netflix is also pretty fun, though it starts to get pretty weird in the later seasons (though I'm feeling hopeful for the new CastleVania series that they just teased). I've also recently enjoyed Chainsaw Man, and am pretty excited for the next season to come out.
I also highly recommend Redline. It's not a series, but a standalone film, and it's one of my favorite movies ever. If you're at all familiar with the F-ZERO games, think of that, but as an anime.
One of my favorites is Black Rock Shooter. It's a bit wierd around the edges and it's presented as this girly slice of life but it's actually deeper and very cool