If you're looking for mainstream recs:

Shounen (male teens / early 20s)
- FMA: Brotherhood - 10/10. About two brothers who commit a horrific mistake and lose their bodies and limbs. So they become alchemists to regain their bodies.
- Hunter x Hunter - 10/10. About a boy named Gon and his journey to become a Hunter (adventurer) to find his father who abandoned him for Hunter work. He teams up with an assassin, an androgynous guy whose entire clan was slaughtered, and an aspiring doctor along the way.
- One Punch Man - I read the manga, but I heard the anime was really good. It's about an average guy who quits his job to become a superhero and climb the rankings and somehow defeats all his foes with one punch.
- Demon Slayer - very popular especially in Japan. It's about a boy who loses his family to demons. Only his sister survived, but cursed with being a demon herself. So he sets off to join the anti-demon military to avenge them.
- Assassination Classroom - it's about an superbeing octopus who blows up half the moon and threatens to destroy the world if some Japanese middle school students he teaches can't kill him within a year.
- Gurren Lagann - also mecha. People now live underground because a brutal dictator forced them to and because surviving on the surface is too dangerous. However, two teens uncover some robot that lets them access the surface.
- Jojo - various arcs involving the Joestar family across generations ranging from fighting vampires, buff superbeings, serial killers, the mafia, prison priests, and other hijinks.
- Samurai Champloo - teenage girl meets up with two rounin (masterless samurais) who help her find her samurai father.

Seinen (male, mid-20s+)
- Attack on Titan - horrific tall creatures bust through the village's walls and start slaughtering and eating everyone. MC's mother is eaten and his life is turned upside down, and he joins the army to exact vengeance.
- Mirai Nikki - a bunch of random people are given phones that can display text messages 90 days in the future and are all involved in a survival game. Whoever wins becomes the next God and can stop the Apocalypse. MC is a male teen who has to survive with this psychotic classmate deeply in love with him.
- Akame ga Kill! - farmer teen heads to the empire capital to earn a living and send money back to his village. But he soon realizes the empire is evil and joins a band of assassins working to undermine the empire.
- Kiseijuu - alien parasites take over Earth, hijack people's brains, and eat humans. Some teen boy also gets half-infected, retains his consciousness, and fights against the other aliens to protect society.
- Trigun - a fugitive, infamous for his proficiency with firearms, has a huge bounty on his head who is hunted down by a couple of investigators.
- Kill la Kill - high school girl's father is murdered by someone she suspects attends some high school academy specializing in fighting with clothes. So she transfers there with the goal of finding the killer and avenging her father.
- Ajin - a very small fraction of humans have evolved to become immortal. Once they're found, governments abduct them for horrific experiments. Some high school boy learns he is one and goes on the run from the government.
- Madoka Magica - middle school girls make contracts with a fluffy animal to fulfill one life wish in return to be enlisted as magical girls to defend the world from demons.
- Darker Than Black - some black portals appeared all over the world that somehow gave people special powers called Contractors. Some Chinese contractor guy with a history comes to Japan under his assassin's guild for work, but he's got other plans.
- Fate Stay Night series - battle royale survival game involving regular people called summoners and their servants from a list of legendary heroes. Some Japanese high school teens get caught up in it. Also, King Arthur is a woman for some reason.
- Goblin Slayer - goblins massacre some guy's family and village through rape and death. He becomes a skilled adventurer who only specializes in killing goblins to avenge his family. Along the way, he meets a priestess, an elf, a lizardman, and a dwarf to join him to kill goblins.
- Gungrave - story about two men living a life of poverty who decide to join the mafia to make a better life. Hijinks ensue and suddenly technology has evolved to allowed for people to transform into monsters and supernatural battles.

Psychological
- Death Note - high schooler gets a notebook where he can write any name and they'll die from a heart attack in a minute. He uses this to kill criminals until the police had enough of looking bad and his unethical conduct and tries to track him down.
- Steins;Gate - some unemployed guy with a mad interest in science with his sidekick hacker friend and childhood romance hang out in some shack where he invents a way to send emails to the past to affect the present. Time travel hijinks and government conspiracies ensue.
- Monster - a Japanese neurosurgeon saves a European boy's life. But the boy he saved is coincidentally a psychopath and starts murdering people all over Europe. So he embarks on a journey to find and stop him.
- Neon Genesis Evangelion (also mecha) - some middle school boy is forced by his father to pilot a robot to protect Earth from giant demons.

Political / Mecha
- Code Geass - some insanely smart high school boy gains the power to mind control people. He uses this power to learn who killed his mom and blinded his sister. Oh, and he's the son of the British emperor whom he deeply hates. Also, this Britsh Empire annexed Japan and are racist to them. So in order to find out, he joins the Japanese revolutionary army to fight back.
- Gundam 00, Seed, Wing, IBO, Origin - teenage men or young adults somehow gain access to piloting a giant robot to fight against some other political side with their own giant robots. Usually, it's some space colony declaring independence from Earth. Space opera, huge robot battles, and yelling from inside cockpits ensue.
- Eureka 7 - some teenage boy gains access to a robot that can surf in the sky. He gets caught up in some political stuff and joins a rebel squad and becomes insanely attracted to an enigmatic blue-haired girl who is also the squad's ace pilot.
- Bokurano - some elementary and middle school kids in summer camp get recruited by some giant robot pilot against their wills to pilot the suit to fight against other giant robots trying to destroy their planet.

Comedy
- Grand Blue - some university students get drunk every day and sometimes they go diving. Bunch of romcom stuff too.
- Daily Life of High School Boys - some high school boys doing some dumb shit to entertain themselves
- Saiki-kun - an esper has to navigate high school life with a bunch of weirdos using his psychic powers with oddly specific rules.
- Cromartie High School - some high school boys doing dumb shit in a nonsensical school
- Baka to Test - some high school is organized based on test scores and the lowest ranking class aims to climb back to the top with its misfits through RPG battles
- GTO - a Japanese redneck becomes a middle school teacher and tries to flirt with another female teacher and students' moms while trying to work out their issues and causing constant grief for his asshole boss
- Hinamatsuri - some yakuza guy adopts an emotionless esper girl who causes him constant grief
- Is This a Zombie? - some high school guy who became a zombie gets involved with a bunch of hijinks including his necromancer, a magical girl, and a vampire ninja to fight against random demons

Music
- Nodame Cantabile - some piano gifted airhead tries to coerce an asshole conductor by bonding through classical music
- Sakamichi no Apollon - some high schoolers bond over their love of jazz

Romcom
- Kaguya-sama: Love Is War - two high schoolers like each other try to coerce the other one into confessing to the other because they fetishized their superiority complexes
- Ouran High School Host Club - some poor high school girl accidentally breaks a high school club's antique vase and has to pay back her debt by joining her club and crossdressing and working as a platonic gigolo

Romance Drama
- Toradora! - two neighbouring high school students help wingmen each other so that they could hook up with their crushes
- Clannad - big-eyed high school girls get their issues solved by a depressed male classmate

Sports
- Haikyuu! - some short guy joins a volleyball team

Non-JP / Western
- Avatar - some Tibetan kid got frozen in solid ice for 100 years, thaws out, and learns his entire clan was slaughtered by Imperial Japan, so he goes on a quest to master all four elements with two Eskimos so he can take on the evil empire's leader
- Castlevania - a woman fell in love with Dracula because of their mutual love for science. She introduced his advanced tech to Dark Ages Europe and priests burned her for being a witch. Dracula learns of this and vows to genocide humanity. A vampire hunter teams up with a Romani magician and Dracula's half-vampire son to stop Dracula.
- Blood for Zeus - Zeus cheated on Hera with a human and a son is born. That son grows up and comes to terms with his destiny to protect the world and Gods from demons.

EDIT: Romani, not gypsy. Sorry.