Manga

These are grouped by author.

Mostly known as the author of Dragon Ball, the work of his that I most like is Dr Slump: humorous and surreal story of an inventor who builds sentient robots with the most absurd issues. Toriyama's humour still echoes in recent Japanese works.

TAKAHASHI Rumiko

You probably know her for Inu Yasha and Ranma½, I would suggest instead reading her short stories (Mermaid Saga, One-pound Gospel, Rumic Theater) and her first long series, Urusei Yatsura: probably the first example of harem story / sudden girlfriend appearance, funny and sweet.

YUKINOBU Hoshino

Read everything he's written. 2001 Nights and the sort-of sequel, 2001+5, are great examples of golden-age hard SF seen from a different perspective.

TEZUKA Osamu

The god of manga. Japanese popular culture would be very different without him. Favourites of mine:

the Phoenix saga, an exploration of life, death, rebirth, across time and space.
Message to Adolph, story of three men named Adolph through WW2
Princess Knight, adventures of the two-hearted princess who lives as a man (more or less, it's complicated)
MW: chemical weapons, serial killers, gay relationships
Ode to Kirihito

But seriously, read as much Tezuka as you can get your hands on. Warning: some characters are depicted in a way that looks quite racist today, but was "common" at the time. Make of that what you will.

OTOMO Katsuhiro

Writer and director, justly famous for Akira. The movie is good, but the manga has much more depth and breadth.

SHIROW Masamune

Ghost in the Shell, obviously: the manga has similar themes to the movie, but it's filled with jokes
Appleseed, slow and complicated story of political manoeuvres and discrimination
Dominion, not-very-competent militarised police
Orion fantasy / science fiction inspired by the Japanese myths of creation

In general, Shirow puts plenty of jokes in his works, and his obsession with guns and tanks &c may be off-putting, but there's serious thoughts underneath it all.

MIYAZAKI Hayao

Yes, one of the founders of Studio Ghibli also wrote manga. His masterwork is Nausicaa of the Valley of Wind, a story of epic proportions about war, friendship, environmentalism, and the future of humanity. The movie of the same name is a pale shadow of the manga.

KEI Toume

Her manga are dark and melancholy. Kurogane, Hitsuji no Uta (Lament of the Lamb), Hatsukanezumi no Jikan (Time of the Mice).

KITOU Mohiro

Raw and bloody. Narutaro, Bokura No, Wings of Vendemiaire

HAGIO Moto

One of the founders of shoujo manga. Only work of hers I've read so far is They Were Eleven, '70 SF at its best.

TSUDA Masami

His and Her Circumstances is a very nice romance story: the characters talk to each other! And when the obvious couple get together, they stay together and work through their problems!

TAKAHASHI Shin

Author of one of the saddest stories I've ever read, Saishou Heiki Kanojo (a.k.a. "she, the ultimate weapon").

YAMAGUCHI Takayuki

Kakugo no Susume half-serious, half-parody post-apocalyptic fighting story: cursed power armours built by Imperial Japan, mutated insects and monstrous plants used to frame shoujo-style panels, Buddhist precepts, and a tremendously queer antagonist.

IWAAKI Hitoshi

Kiseiju: the aliens have invaded, and they look just like us. Is pacific co-existence possible?

Also Historie, but I haven't read it yet.

CHOUYA Koyama

Space Brothers, a love letter to space exploration and the people who make it possible.

UEO Hisamitsu

The Qualia of Purple mind-bending story of a girl who sees people as robots, and of her friend who wants to protect her.

Puyo, inspired by TANIGAWA Nagaru

Tanigawa wrote the Haruhi Suzumiya series of light novels, later adapted to manga and anime. I think that the best manga set in that universe is The Disappearance of Nagato Yuki-chan (warning: spoilers for The Disappearance of Haruhi Suzumiya)

URASAWA Naoki

Pluto a re-interpretation of Tezuka's Mighty Atom (Astro Boy) with a more modern, realistic look
Monster killer psychopaths
20th Century Boys can childhood make-believe spell out the end of the world? can they help prevent it?

MURAYAMA Kei

A Centaur's Worries starts as a school comedy with centaurs, angels, sheep-people, mermaids. It then mixes in SF elements, conspiracy theories, international politics, and I'm still not sure where it's going.

Chiho Saito and Kunihiko Ikuhara

Revolutionary Girl Utena the manga has a different story than the anime; I still have to watch the anime.

ASHINANO Hitoshi

Yokohama Kaidashi Kikou slice of life of an android running a country café in a world after a global flood, very slow and poetic, beautifully drawn.

Anime

Series

Haibane Renmei: maybe a metaphor for Purgatory, maybe for socialism, maybe just a story to learn to forgive one's self, this series is beautiful
Top wo Nerae - Gunbuster: One of the first works by Gainax, with character design by MIKIMOTO Haruhiko and direction by ANNO Hideaki. Giant robots, teen angst, special relativity. The sequel, Daibuster, is also remarkable for keeping up with the scale of the original.
Cowboy Bebop: See my review.
FLCL: The start of the "surreal" period of Gainax. A weaponised guitar, a Vespa, double-entendres, unusual relationships.
The Melancholy of Haruhi Suzumiya: “I have no interest in ordinary humans. If there are any aliens, beings from the future, or super-humans, come see me. That is all!” Between zany and profound, this is one of the most intriguing series I've even seen. Originally broadcast intentionally out of order, I think it works better watched in that order than in the straight internally chronological one.
Future Boy Conan: Early work by, among others, MIYAZAKI Hayao and TAKAHATA Isao. Many Ghibli films will use themes and character details from this series.
Panty & Stocking with Garterbelt: See my review.
Durarara!!: See my review.
Madoka Magica: See my review.
Gatchaman Crowds: See my review.
Lupin III: The Woman Called Fujiko Mine: See my review.
Nadia - Secret of Blue Water: See my review.
Kyousougiga: See my review.
Sora no Woto: Young girls crew a military base in not-quite-Switzerland. War, relationship between military and civilians, trumpets, old myths.
Dai Mahou Touge: This is not a standard magical girl story. If Punie can't win with magic, she'll win with uncontrolled brutality.
Dennou Coil: Intrusive virtual reality, hacking, friendship. Oh, and cute girls with glasses.
Eve no Jikan: Exploration of a society with humanoid, Asimov-style robots. Also watch the movie and the related short Mizu no Kotoba.
Macross: If you've only seen Robotech, you owe it to yourself to see the original version of one of the three works that were mashed together for the US rewrite. The best summary of Macross is "humans win a war against violent aliens by singing at them".
Hibike! Euphonium: High school brass / marching band club, working hard to pass the selections toward the national contest. Good music, good portrayal of the amount of work involved in learning to play musical instruments, and a very sweet romance.
Shin Seiki Evangelion: The series that re-booted the "giant robots" genre. Gory, messy, with an ending that still has people arguing after twenty years and several animated reiterations.
Kanata no Astra: Relentlessly optimistic space adventure. At the beginning it feels pretty similar to "They Were Eleven" by Hagio MOTO, but then it goes in a very different direction.
Devilman: Crybaby: To the best of my knowledge, Crybaby is the first faithful and complete adaptation of the Devilman manga.
Flip Flappers: Magical girls, dimension-hopping, Akira-like ending.
Carole & Tuesday: Rich girl runs away from home, meets poor girl, they become a musical sensation. Also, fight fascism with songs.
ODDTAXI: A city populated by anthropomorphic animals, a taxi driver, the surprisingly interwoven lives of his clients. It reminded me of Durarara!! both for the semi-fantastical urban setting, and for the writers' ability to tie many plot lines together.

Movies

All movies by Studio Ghibli: From Nausicaä of the Valley of Wind (technically pre-dating the founding of Ghibli) to When Marnie was There, via Totoro, Pon Poko, Kiki, Porco Rosso, Mononoke, Spirited Away, Howl's Moving Castle, Arrietty, The Wind Rises, Princess Kaguya. Ghibli has been essentially the benchmark of excellence in animation for thirty years.
Avalon: Live action movie by OSHII Mamoru, about immersive games, subjective realities, and Arthurian myths.
Ghost in the Shell: Oshii's best known work, keeping close to the events narrated in the original manga but injecting a very personal meaning.
Wolf Children: Growing up, responsibilities, following one's true calling.
Garden of Words: See my review.
009 re:Cyborg: See my review.
The Disappearance of Haruhi Suzumiya: See my review. This movie is set after the second season of the series.
Pale Cocoon: By the same author as Eve no Jikan. A story about re-interpreting the present by re-discovering the past.
Hoshi no Koe: SHINKAI Makoto's big debut, about very-long-distance relationships.
Urusei Yatsura: Beautiful Dreamer: Oshii's first movie, all his trademarks can bee seen already.
Puni Puni Poemi: Over-the-top absurdist magical girl. If you thought that Excel Saga was too long, and not surreal enough.
Miss Hokusai: Well made historical about the life of the famous painter Hokusai from the point of view of his daughter Oei

Anime & Manga I like

Manga

Anime

Series

Movies