I'm going to answer your question in a slightly different way by recommending shows that I've found instructive about Japanese culture.
As a quick introduction to the representation of Japanese culture in anime and manga, I recommend
School Rumble. Consider it a "crash course" that covers the key icons of high school life, all in a very funny package. To name just a few, you'll see
school uniforms,
bento boxes from girls courting boys, festivals and athletic competitions, the mandatory
beach and
hot springs episodes, and children who run their own households without any parents around.
I would hope that the last of these is much more common in anime than in real-life!
The supernatural often plays a role in anime and manga. Here I'd recommend watching two very different shows,
Mononoke and the more light-hearted
Natsume Yuujinchou. Both have an array of serious and comedic portrayals of demons and other supernatural beings.
Another show you might sample is
Genji Monogatari Sennenki, an animated re-telling of the
Tale of Genji by the Lady Murasaki. The
Tale is considered to be the world's first novel in any language and was written a thousand years ago.
Finally I'll suggest the movie,
Summer Wars, and the currently-running series
Usagi Drop, for their portrayals of extended family life in Japan.