1: Toradora!
Toradora!'s story begins with the male protagonist Ryuuji Takasu who, despite his gentle personality, has a natural outer appearance which makes him look like an intimidating delinquent. Therefore, he is utterly hopeless about his chances of getting a girlfriend anytime soon, and does not have many close friends either.
After being greeted by his hungover mother in the morning, Ryuuji goes to school and is happy to find that he gets to be in the same class as his best friend, YÅ«saku Kitamura, and crush, Minori Kushieda. However, it is then that he unexpectedly knocks into "the school's most dangerous animal of the highest risk level" — Taiga Aisaka — who just happens to also be in his class.
Despite the quite obvious anime stereotypes in this show's characters, Toradora is among the best anime ever made. The author manages to take the stereotypes and run with them, making even the most mundane and common ones seem almost deep and psychological. The love story is cute and funny, the main characters' development is incredible and fascinating. Great artwork, fun music, a fast-paced ever-changing story line, and well-fleshed supporting characters make this an anime you'll never forget and always treasure.
#2: Death Note
Yagami Light is an ace student with great prospects, who's bored out of his mind. One day he finds the "Death Note": a notebook from the realm of the Death Gods, with the power to kill people in any way he desires. With the Death Note in hand, Light decides to create his perfect world, without crime or criminals. However, when criminals start dropping dead one by one, the authorites send the legendary detective L to track down the killer, and a battle of wits, deception and logic ensues...
Death Note is thrilling, fascinating, fast-paced, and highly intelligent. The plot is thought-provoking, dark, and never ceases to amaze, and the show seems inexplicably to stay fresh, new, and exciting each time you rewatch it.
#3: 12 Kingdoms (alternative title: Juuni Kokki)
A Japanese high-school girl is mystically transported to another world where she discovers her true fate; largely a political drama.
the story is about Yoko Nakajima, a girl who felt compelled to keep up an image of being a model student both in school and also at home.
One day, a man named Keiki showed up suddenly and recognised Yoko as his master. From that moment on, her life began to change dramatically.
Along with a couple of her classmates, she was taken by Keiki into a fantasy world made up of 12 Kingdoms. There, she became lost and separated from Keiki. Hunted by demons and mistreated by everyone (even her own classmates), she becomes strong through the ordeals she faced in order to survive in the world of the 12 Kingdoms and uncovers the destiny that lay in store for her.
This anime is GREAT. Character development is awesome, art is good, the only lacking parts are the animation and the music (well, the music is O.K.). However, the flaws are tremendously overshadowed by the general perfection of the storyline, the characters, the psychological intrigue... It may seem slow-paced and it does require a lot of intellectual energy but you will not be disappointed; each episode is fascinating and leaves you feeling very satisfied.
These shows are THE BEST, MUST-WATCH anime. !!!
Enjoy watching!! :D
~Edit~ Here are some other very good shows that aren't quite THE best but are very much worth watching. I hope this helps!
Mononoke
Planetes
Baccano!
Nodame Cantabile
Monster
Dennou Coil
Maison Ikkoku
Darker than Black
Ghost in the Shell Stand Alone Complex
Cowboy Bebop
Kimi ni Todoke
Bokura ga Ita
Kanon
Cross Game
FLCL
Neon Genesis Evangelion
Fullmetal Alchemist
Paranoia Agent
Gankutsuou - The Count of Monte Cristo
Serial Experiments Lain
Samurai Champloo
Trigun
Fruits Basket
Azumanga Daioh
Lucky Star
Cardcaptor Sakura
Kino no Tabi
ahhhh there are so so so many more great ones but I think these'll tide you over for now! Haha... Sorry if it's too long...