Here is a long list of some of the manga that I've enjoyed from different genres. Most have anime shows/movies/OVAs attached to them, though I can't vouch for many (you're welcome to watch them though). Take what you will from this list and hopefully you'll find a couple you like:
(Predominantly) Action:
Ongoing
-
Ansatsu Kyoushitsu (Assassination Classroom): probably the closest to your type of manga on this whole list (comedic action)
-
Dogs: Bullets & Carnage: there's a prequel (
Dogs: Stray Dogs Howling in the Dark) that's already complete. I suggest reading that first.
-
Pandora Hearts: I've stopped reading this for a while, but it was really good up to the point I stopped (soon to be finishing from what I've heard)
Complete
-
Claymore: fantasy, adventure; the plot development is a bit wonky, but it's good nonetheless
-
Donten ni Warau: there is now an ongoing sequel
-
Kiseijuu (Parasyte): sci-fi; a bit old (1989-1995), but obviously popular enough to garner an anime after all these years
-
Mugen no Juunin (Blade of the Immortal)
-
Tokyo Ghoul: there's also the sequel, which is ongoing.
-
Tsubasa -RESERVoir CHRoNiCLE-
Comedy
Ongoing
-
Nichijou: who doesn't want to see a principal wrestle a deer? Mix of 4-koma and normal panels
-
Yotsubato! (Yotsuba&!): it's also a cute slice of life. Not much of a story, but enjoyable nonetheless
Complete
-
Azumanga Daioh: 4-koma about the randomness of five high school girls. Yeah, don't expect a story.
-
Kami Nomi zo Shiru Sekai (The World God Only Knows): parody and satire
-
Sayonara Zetsubou-sensei: dark satire. Should you expect a story from this? No, not really.
Drama, Romance, School, Slice of Life: (One, two, or all of these genres because I'm a bit lazy to separate them and some of them fall in many)
Ongoing
-
Chihayafuru
-
Kimi ni Todoke (From Me to You)
-
Nana: this is on hiatus...
-
orange: your typical shoujo characters, which irritates me, but it deals with time travel, parallel worlds, and trying to make the right decisions to avoid an accident with the main male character.
Complete
-
Bakuman.: a manga about becoming mangaka created by the creators of
Death Note
-
Bokura ga Ita (We Were There): starts off like any other light-hearted, fluffy shoujo manga, but it gets very mature
-
Hachimitsu to Clover (Honey & Clover): drama, romance, slice of life all in a university setting
-
Hanazakari no Kimitachi e (Hana-Kimi): first gender-bender I read
-
Kaichou wa Maid-sama!
-
Nodame Cantabile: every version of this is amazing. Manga, live-action drama, and the anime.
-
Ouran High School Host Club
-
Sakamichi no Apollon (Kids on the Slope)
Psychological:
Complete
-
Death Note: psychological and supernatural
-
Liar Game: recently completed on the 22nd
-
Monster: this is intense and lengthy stuff
-
Oyasumi Punpun (Goodnight Punpun): excuse the name. It's a coming of age story that I find to be very psychologically focused.
Sports:
Ongoing
-
Haikyuu!: volleyball and comedy
-
Yowamushi Pedal: cycling (sounds lame, I know, but it's some good stuff)
Complete
-
Eyeshield 21: American football with tons of comedy
-
Kokou no Hito (The Climber): rock-climbing
-
Kuroko no Basuke (The Basketball Which Kuroko Plays): basketball and comedy
-
One Outs: baseball; a bit more psychologically-focused than sports-focused though. No one has fully scanlated the manga, but you can watch the anime for the first part, then read the rest.
-
Ping Pong: self-explanatory
-
Slam Dunk: the old basketball classic
Supernatural: (I know some stuff could fall into the supernatural category, like Tokyo Ghoul and Death Note, but they weren't the first genre that came to mind for me. Others will say differently)
Ongoing
-
Kuroshitsuji (Black Butler): Victorian-era, supernatural mystery with some comedy; didn't like the anime except for
Book of Circus 'cause that was an arc straight from the manga
-
Natsume Yuujinchou (Natsume's Book of Friends): supernatural and slice of life. Peaceful manga and anime. I also liked
Hotarubi no Mori e from the same author.
Complete
-
Mushishi: calming, life, and nature. That's all you really need to describe it. And of course "Mushi". Both the manga and anime are great.
-
xxxHOLiC: the thing about this and Tsubasa is that they're so closely intertwined that you have to read them together to know everything that's going on. A bit of a hassle, but very much worth it.
Misc.: (don't know where to fit it)
-
Hakushaku Cain (Count Cain) and
Godchild: a dark and twisted Gothic horror based in the Victorian-era. Comparable to Kuroshitsuji, but less supernatural.
