This is a list of Anime and Manga I personally like, and will of course add anything else that I run across that I like and recommend. Anime and manga are huge in Japanese culture. It’s pretty much everywhere (within reason).
But they are great learning platforms to really start understanding Japanese, and Japanese culture. Especially if you read manga, if you want to know how the culture as a whole speaks and acts, pick up a current manga and read it and you’ll see pretty much exactly how every day life is lived (without the robot explosions, monsters and evil demons).
Welcome to the NHK – The life of a hikikomori conspiracy theorist
Princess Jellyfish – A look into the Japanese gay and otaku culture in Japan.