Some (varied) recommendations:
Fate/stay night - [unlimited Blade Works] - Ufotable's first adaptation of a route from the Fate/stay night visual novel, it keeps the general aesthetics of Fate/Zero but changes the tone accordingly. As an anime first viewer, I really enjoyed it as a gigantic step above Studio DEEN's 2006 Fate/stay night and, though Fate/Zero is overall better, this is a worthy continuation. Next year there will be a trilogy of movies on Fate/stay night - Heaven's Feel, also animated by ufotable. Here's a quick synopsis:
The Holy Grail War is a battle royale among seven magi who serve as Masters. Masters, through the use of the command seals they are given when they enter the war, command Heroic Spirits known as Servants to fight for them in battle. In the Fifth Holy Grail War, Rin Tohsaka is among the magi entering the competition. With her Servant, she hopes to obtain the ultimate prize—the Holy Grail, a magical artifact capable of granting its wielder any wish. Another magus, Shirou Emiya, wants to become a hero of justice and prevent suffering, but he'll have to question his ideals when unwittingly thrown into a deadly battle royale. Unlimited Blade Works can be seen as an alternate version of the Fate route adapted in the 2006 anime, but doesn't need to be seen to understand UBW. It is recommended to watch Fate/Zero before UBW. The difference between Fate and UBW is the focus changes from the relationship between Emiya and his Servant to that of Emiya and Tohsaka, as well as her Servant.
Recommended for: great action scenes, stellar animation, focus on characters.
Not recommended if: if you watched Fate/Zero and it set your expectations for a follow up to be the same in tone, UBW might share F/Z's general atmosphere (if a bit less tense) but it is more idealistic than F/Z and focuses less on the Holy Grail War and more on the main characters' development.
JoJo's Bizarre Adventure - David Production followed up on the 2012 JJBA with an incredibly faithful 48 episode adaptation of Stardust Crusaders and an ongoing also very faithful adaptation of Diamond is Unbreakable that will run for 39 episodes. Requires watching the 2012 JJBA beforehand.
Stardust Crusaders - In a Japanese jail sits 17-year-old Jotaro Kujo: punk, fighter, delinquent...and possessed by a force beyond his control. Around the world, evil spirits are awakening: "Stands," monstrous invisible creatures which give their bearers incredible powers. To save his mother's life, Jotaro must and travel around the world to Cairo, Egypt, where a hundred-year-old vampire thirsts for the blood of his family. But the road is long, and an army of evil Stand Users wait to kill JoJo and his friends...
Diamond is Unbreakable - The year is 1999. Morioh, a normally quiet and peaceful town, has recently become a hotbed of strange activity. Jotaro Kujo, now a marine biologist, heads to the mysterious town to meet Josuke Higashikata. Allied together with two of Josuke's friends, the group dedicates themselves to investigating recent disappearances and other suspicious occurrences within Morioh. Aided by the power of Stands, the four men will encounter danger at every street corner, as it is up to them to unravel the town's secrets, before another occurs.
I personally recommend watching Diamond is Unbreakable if you enjoyed the 2012 JoJo even if you disliked Stardust Crusaders. While both SC and DiU follow a sort of enemy of the week structure, DiU is infinitely more creative in the way it's approached and presented. SC focuses on the journey, DiU focuses on the characters' daily lives and how they use their Stands in that regard. Regardless, both are very good series in their own right and I recommend both.
Recommended for: great action scenes, fantastic animation, brilliant over-the-top style
Not recommended if: you dislike long monster of the week series, dislike a loud, boisterous over-the-top style, consider underdeveloped characters a dealbreaker, don't like gore.
Recommended for: creative action scenes, fantastic animation, brilliant over-the-top style, creative use of typical shonen traits in a slice of life environment.
Not recommended if: you dislike a loud, boisterous over-the-top style, don't like gore.
If you are interested in DiU but not SC, I recommend watching episodes 1-5, 23-26 and 40-48 (10-12, 21, 22, 27-29, 36 and 37 added if you care about Polnareff.) to be familiar with what went on (and frankly the last 9 episodes are glorious).
Food Wars - A great comedy ecchi cooking anime that is surprisingly very tense. 24 episodes aired last year with another season coming out next week.
Ever since he was a child, fifteen-year-old Souma Yukihira has helped his father by working as the sous chef in the restaurant his father runs and owns. Throughout the years, Souma developed a passion for entertaining his customers with his creative, skilled, and daring culinary creations. His dream is to someday own his family's restaurant as its head chef.
Warning: first episode spoiler
Recommended for: hilarious comedy, surprisingly tense moments, over-the-top style, gorgeous plates.
Not recommended if: ecchi is a turn-off, don't like an over-the-top approach to food-tasting.
One Punch Man - a subversive superhero comedy show. Aired last fall for 12 episodes, has an OVA and 6 12-minute specials.
The seemingly ordinary and unimpressive Saitama has a rather unique hobby: being a hero. In order to pursue his childhood dream, he trained relentlessly for three years—and lost all of his hair in the process. Now, Saitama is incredibly powerful, so much so that no enemy is able to defeat him in battle. In fact, all it takes to defeat evildoers with just one punch has led to an unexpected problem—he is no longer able to enjoy the thrill of battling and has become quite bored.
Recommended for: fantastic action, gorgeous animation, excellent comedy, interesting characters, different approach to the superhero genre.
Not recommended if: I really can't think of a warning other than a bit of gore and a small amount of fanservice.
Shimoneta: A Boring World Where the Concept of Dirty Jokes Doesn't Exist - A hilarious ecchi comedy that also serves as a clever criticism on censorship in Japan. Aired for 12 episodes last summer.
With the introduction of strict new morality laws, Japan has become a nation cleansed of all that is obscene and impure. By monitoring citizens using special devices worn around their necks, authorities have taken extreme measures to ensure that society remains chaste. In this world of sexual suppression, Tanukichi Okuma—son of an infamous terrorist who opposed the chastity laws—has just entered high school, offering his help to the student council in order to get close to president Anna Nishikinomiya, his childhood friend and crush. Little does he know that the vice president Ayame Kajou has a secret identity: Blue Snow, a masked criminal dedicated to spreading lewd material amongst the sheltered public—and Tanukichi has caught the girl's interest due to his father's notoriety.
Recommended for: great comedy, social commentary, interesting characters.
Not recommended if: ecchi is a turn-off.
ERASED - A murder mystery thriller with time travel from winter of this year. Aired for 12 episodes.
When tragedy is about to strike, Satoru Fujinuma finds himself sent back several minutes before the accident occurs. The detached, 29-year-old manga artist has taken advantage of this powerful yet mysterious phenomenon, which he calls "Revival," to save many lives. However, when he is wrongfully accused of murdering someone close to him, Satoru is sent back to the past once again, but this time to 1988, 18 years in the past. Soon, he realizes that the murder may be connected to the abduction and killing of one of his classmates, the solitary and mysterious Kayo Hinazuki, that took place when he was a child. This is his chance to make things right.
Recommended for: gripping mystery, well-developed characters, fantastic atmosphere, brilliant directing, different approach to time travel.
Not recommended for: I can't think of anything that could be a turn-off.
KonoSuba: God's Blessing on this Wonderful World! - a hilarious ecchi comedy series that subverts common tropes in the "NEET gets transported to RPG world" wish fulfillment type of story. Aired for 10 episodes last winter, with an OVA coming out soon and a second season in production.
Warning: I recommend not reading the synopsis, this spoils the events of the first episode and they really have to be seen unspoiled for the best experience.
Recommended for: hilarious comedy, subversive characters, unique take on a tripe format.
Not recommended if: its somewhat wonky animation turns you off, you are a fan of otaku wish fulfillment anime. It can still be enjoyed if you like it but it's not what you'd want if all you wanted was a wish fulfillment story.
Shouwa Genroku Rakugo Shinjuu (Shouwa and Genroku Era - A Lover's Suicide Through Rakugo) - a gripping historical drama about the Japanese art of rakugo. Aired for 13 episodes (1st is 40 minutes long) last winter with a second season in production.
Yotarou is a former yakuza member fresh out of prison and fixated on just one thing: rather than return to a life of crime, the young man aspires to take to the stage of Rakugo, a traditional Japanese form of comedic storytelling. Inspired during his incarceration by the performance of distinguished practitioner Yakumo Yuurakutei, he sets his mind on meeting the man who changed his life. After hearing Yotarou's desperate appeal for his mentorship, Yakumo is left with no choice but to accept his very first apprentice. As he eagerly begins his training, Yotarou meets Konatsu, an abrasive young woman who has been under Yakumo's care ever since her beloved father Sukeroku Yuurakutei, another prolific Rakugo performer, passed away. Through her hidden passion, Yotarou is drawn to Sukeroku's unique style of Rakugo despite learning under contrasting techniques. Upon seeing this, old memories and feelings return to Yakumo who reminisces about a much earlier time when he made a promise with his greatest rival.
Shouwa Genroku Rakugo Shinjuu is a story set in both the past and present, depicting the art of Rakugo, the relationships it creates, and the lives and hearts of those dedicated to keeping the unique form of storytelling alive.
Recommended for: Great characters, brilliant storytelling, great directing, gripping drama, great insight into Japanese culture, great animation (it's a DEEN anime with great animation!).
Not recommended if: you dislike slower, more talky series. There's great conflict in this one but it goes through a lot of (beautifully written) dialogue.
Re:ZERO -Starting Life in Another World- - Like KonoSuba, a NEET-in-RPG-World story with a twist. While this one plays things more straight, the very charismatic protagonist and gripping plot make it stand apart. Airing this season for 26 episodes.
Like with KonoSuba, I advise against reading the synopsis but more for plot reasons than comedy reasons.
Recommended for: fun characters, gripping plot.
Not recommended if: you dislike blend comedy-serious action series. The series pulls it off rather well, though.
Kabaneri of the Iron Fortress - a dark steampunk zombie apocalypse story airing this season for 12 episodes (ends next week). Bears striking similarities to Attack on Titan due to sharing production teams. I haven't seen AoT yet so I can't make a proper comparison.
As the world is in the middle of an industrial revolution, a monster appears that cannot be defeated unless its heart, which is protected by a layer of iron, is pierced. By infecting humans with its bite, the monster can create aggressive and undead creatures known as Kabane. On the island Hinomoto, located in the far east, people have built stations to shelter themselves from these creatures. People access the station, as well as transport wares between them, with the help of a locomotive running on steam, called Hayajiro. Ikoma, a boy who lives in the Aragane station and helps to build Hayajiro, creates his own weapon called Tsuranukizutsu in order to defeat the creatures. One day, as he waits for an opportunity to use his weapon, he meets a girl named Mumei, who is excused from the mandatory Kabane inspection. During the night, Ikoma meets Mumei again as he sees Hayajiro going out of control. The staff on the locomotive has turned into the creatures. The station, now under attack by Kabane, is the opportunity Ikoma has been looking for.
Recommended for: well-realized gritty zombie apocalypse story, stunning art and animation, good action.
Not recommended if: you dislike zombie stories played straight. While this takes place in a steampunk setting aboard trains, it is after all a zombie apocalypse. It has enough things both plot and style-wise to set it apart from other stories in that vein, though.
Other series I've heard good things about but haven't seen myself to recommend firsthand include Ushio to Tora, Your Lie in April, Plastic Memories and Himouto! Umaru-chan.