Spoilers for various anime's. I'll try not to give away anything that isn't in the first episode.
It's well known that there are MANY, MANY, MANY anime shows (and even more manga) and I certainly have not watched even a quarter of them but out of those I have watched I'm here to give you the top five (IN MY OPINION) if you want to add anything to the list feel free to put them in the comments.
Now Orange is good, it's great in fact and it's a lot like Clannad in that it's school kids dealing with regular life but with a touch of mysticism with a touch of time travel possibly?
Now the characters are people you can get invested in from episode 1, the only criticism I have is that the MC can be a bit stupid sometimes, to the extent that it can get annoying but it's only 12 episodes so I'm sure you'll manage.
The art style is very unique as well.
Clannad is very special because while it's mostly just story of a boy befriending a friendless girl and helping her with her dream of making a drama club at school it's got an unearthly other world for a few minutes every episode (which at first seems totally random and weird and boring but totally makes sense at the end) as well as a ghost wondering the school trying to get people to go to her sisters wedding.
Not only is it hilarious with funny characters and some kick ass females but also at it's heart it's about teenagers at school just trying to make something of themselves so they can have a good life, a good education and get into their next stages of learning.
I personally think that anyone with a heart should like this. BTW there's a sequel called Clannad After Story
6. Death Note
Death Note is an amazing 35 episode anime and that's why it's number 4th on this list. The main character, Light, is complex and intelligent, trying to be the hero whilst simultaneously being the villain. He starts off just trying to rid the world of bad after finding a strange book and ends up in the middle of a full scale witch hunt, helping the Japanese police task force assembled to stop him... find himself. And if that's not good enough then throw in a couple Shinigami an angel of death who loves apples, Misa Misa the 'love interest' of sorts and L the genius detective who's a bit of a weirdo.
It's also battles with the constant argument of death to criminals and whether it should be done and of course the ever growing popularity of (somewhat) vigilantism. At it's core without all the mysticism it's a detective show really.
The plot is strong, the characters strong and the dialogue is strong. So if you like anime go watch it now.
5. Beyond the Boundary | Kyōkai no Kanata
Beyond the Boundary is the 13 episode story of a half-demon and a demon hunter brought together by unexpected turn of events. The imagery (especially in the credits) are beautiful and the plot carries with great characters. It's the perfect mixture of great humour and tear jerking sadness with heart-wrenching back stories and an OTP you will be proud to ship. The story draws you in by episode 1 and makes you sad to see it go by episode 13 (So much so I scoured the internet for a movie for half an hour after I finished it). It's got funny and quirky characters as well as an original take on the world of demons and hunters.
Now some people say it's not Nagomu Torii's best work and I personally wouldn't know, all I know is that it's great and you should totally watch it.
4. Sword Art Online
I have to say for a long time Sword Art Online was my favourite anime and right now it's in my top three personally. The reason it's not top of the list is just because while the heart felt moments are very very heart felt and sincere, sometimes it can just take a while longer then it should.
Sword Art Online tells of a story of a gamer Kirito who gets trapped in a game with 100,000 other people where if you die, you die in real life too. The only way to escape is if someone clears all 100 floors of Aincrad the place where the game is set. They tell so many great stories in only 22 minutes it's unbelievable. The fight scenes are intense and the stakes are high, I'm not going to lie though there are casualties, if you don't want to cry don't watch it especially episode 3 that was so sad. I mean don't get me wrong there are many funny, humorous moments as well like all good shows you need a blend of both light and dark.
And if you do watch it make sure you watch Sword Art Online II. And if you want more Sugu with the girls after SAO make sure to watch the season 1 special.
3. Log Horizon
Log Horizon is what some might call another SAO but they're quit different. Whilst Sword Art's Main Character (MC) is driven by his fighting skills, Log Horizons MC is a strategist. Shiroe finds himself trapped in a game that was going out of style with a bunch of his old game friends and becomes a prominent figure in their new world. Another difference between the two shows is that if you die in the game, you come back but you lose memories of the real world.
Season 1 of Log Horizon was honest to God one of the best pieces of television I've ever watched with just the right amount of tension and 'filler' to work without making me die of boredom and developing their characters. Whilst season 2 wasn't as good, and seemed more filled with filler, introducing wackier characters for comic relief it also introduced huge and game changing events that makes me excited to see more if the author ever gets around to writing it after his 10 month stint in prison for alleged TAX Evasion, which he has since paid back.
2. The Heroic Legend of Arslan|Arslan Senki
I truly love Arslan Senki! The art is great, the battle scenes are amazing and you can tell just from the opening credits that it's a high quality anime.
The first episode is quite different from the rest as there is a time jump inbetween.
In a big way this is a coming of age story, a young boy's struggle to gain the respect of his father and those around him whilst at the same time a story of a Crown Prince at war, battling to save his kingdom from the evading forces that have captured their king and taken over their cities. And at an even deeper level it's a story of acceptance and how those opposed, aren't so different.
Prince Arslan has to save the kingdom and his father whilst also trying to fight against a man who claims to and might just be the rightful heir to the throne, the true Crown Prince of Pars.
Honorary Mentions in no order:
Ouran Highschool Host Club ( It's hilarious)
Fairy Tail (I LOVE THAT SHOW)
Not technically an anime but: Teen Titans (No one hate me please) and don't be disheartened by that disaster of a TV show Teen Titans Go!
Mew Mew Power is actually quite good for a kids anime.
Attack on Titans (It's really good and the reason it didn't rate Top 8 is because I haven't finished watching yet).
AND NOW NUMBER ONE ON MY LIST!
Where can I even begin with Bleach? It's 366 episode of pure gold! A 15 year old boy Ichigo can see the ghosts of the dead people and one day an evil spirit called a hollow attacks his town and his family and of course Soul Reaper Rukia Kuchiki comes to kill that hollow but when Ichigo interferes and Rukia loses her powers for a while Ichigo has to step in and become a substitute Soul Reaper. Unfortunately for them both, substitutes are sort kind of illegal and certain people aren't happy with Rukia. From that point onwards, his life is a mesh of hollows, soul reapers, fighting and homework because of course he needs to go to school to keep his grades up or at least average.
Now you're probably wondering why this one is first, in front of all the others, reason being it's just so great. It's got the intense fight scenes, the feels and enough uncanon ships to have a whole navy! On top of that it's HE-HI-HO-LARIOUS with certain gags that carry throughout the show. All the characters are very different, each fighting their own problems and demons (sometimes quite literally) and this makes them interesting to interact with onscreen.
On top of all this they have some bad ass female characters like Rukia, Orihime, Tasuki, Captain Unohana, Karen, Yuzu, Flash-master Yuruichi, Lieutenant Matsumoto and Captain Sui-feng.