Finish Cowboy Bebop.
Wolf’s Rain is meditative and poignant, also by Madhouse Studio.
Fullmetal Alchemist is wonderful! Read the manga, then watch the first anime (it diverges from the manga at about episode 30 and book 7), then watch FMA: Brotherhood. Don’t bother with the movies. The author, penname Hiromu Arakawa, also does the series Silver Spoon, which is set in an agricultural high school academy in the country region, Hokkaido, and it has taught me a whole lot about farming so that I appreciate it much more than before.
Trigun was written/illustrated by Yasuhiro Nighthow, a Catholic! Vash the Stampede, the main character, is my favorite exemplar of a pro-life hero.
I no longer recommend Fruits Basket because it’s a psychological drama that places all the blame for kids’ problems unproductively on caricatures of parents (a common fault in manga/anime because traumas have to come from somewhere). However, I recommend the manga by the same author, Hoshi wa Utau.
Sergeant Frog is a kids show in Japanese, but if you can find the English dub you will find a hilarious parody.
Kodocha, short for Kodomo no Omocha, is an oldie but definitely a goodie. I read it when I was little. It’s comedy/romance, but trust me when I say there is no comedy like it. Read the manga if you can and watch the anime in English.
Rumiko Takahashi is one of my favorite authors. She did Urusei Yatsura, Maison Ikkoku, Ranma 1/2, and Inuyasha. She also did One Pound Gospel, in which she displays the stunning and insulting ignorance of Christianity common to Japanese culture, so stay away from that one. There are a lot of breasts in her illustrations but honestly it’s just for comedy. Think PG-13 Looney Tunes in the case of Ranma 1/2. Despite the risqué plots and situations she’s much more conservative in attitude toward sexual (aka gender) roles. Right now her series Rinne is ongoing, and although the character Tsubasa is again representative of willful Japanese ignorance of Christianity, it’s still freaking hilarious! I own most of the Inuyasha DVD sets

If you like running gags, Rumiko Takahashi’s works are great. I read the Ranma 1/2, Rinne, Urusei Yatsura, and Maison Ikkoku manga but watch Inuyasha.
Samurai Champloo is a really cool but weird anime. It is pretty offensive, though.
Noragami is an OT-rated supernatural manga with two seasons of anime. I like it a lot
Koe no Katachi/ A Silent Voice is a manga (with an award-winning movie) about a kid who bullied a deaf girl during elementary school and repents and makes friends with her in high school.
Yotsubato/Yotsuba&! is a comedy/slice of life manga-only series by Kiyohiko Azuma, the same author of Azumanga Daioh. It follows the day-to-day activities of Yotsuba, a preschooler who explores life with her neighbors. She’s odd, adorable and so funny you’ll cry.
Finally, the legendary movie director Miyazaki wrote and illustrated a seven-book series on which his movie Nausicaa of the Valley of the Wind was based. It’s reeeeally environmentalist and borderline nihilist, but it’s also beautiful, so just take what’s good and beware of Miyazaki’s prejudices (he was a young radical at the time of writing).
As for shounen…um, Tokyo Ghoul started out as an ok if sickening horror manga but fizzled out without a conclusion.
There’s also a very interesting psychological drama called After School Nightmare that seems to be sexually ambiguous but actually has a twist metaphor plot.