xxxHolic probably taught me more about Japanese culture than any other manga/anime. That series is saturated in it. And it doesn't hurt that it's by CLAMP, who I love.
In terms of series that affected me, I think Sailor Moon is probably one. I guess because of the themes. It has a lot of great themes about love, friendship, sacrifice, and more.
Chobits was like that for me too. I've only read the manga of it though. It definitely has themes relating to how humans interact with technology that are very poignant.
Magic Knight Rayearth is my favorite manga series (and one of my favorite anime series) and I think it has great lessons on just what our own willpower can accomplish, and the dangers of leaving the fate of a country dependent on the will of a single, imperfect person.
One manga series I started reading a while ago but have just recently gotten back to is The Good Witch of the West. (I also watched the 13-episode anime a while ago). That series has a fantasy setting but is also really deep and eerily realistic in terms of all the crazy political intrigue and the world itself which is clearly not what it seems. I've never read a manga before that was like that. Crazy.
Basically, I flat out just love CLAMP. Their series are very well written and drawn, and they have stories that make you think (though maybe less so in lighter series like CLAMP School Detectives and its two sequels). They're my favorite manga-kas besides Naoko Takeuchi (author of Sailor Moon).
On a lighter note, there's also Kitchen Princess, a manga which I recently started getting from the library. I'm learning about all sorts of foods from that one; it helps that they include recipes in the back of each volume!