Genshiken!
I am generally not a big fan of popular anime shows, but I do like Studio Ghibli films, and I enjoyed Tokyo Godfathers. My favorite anime show so far has been Genshiken, because it is so solidly based in reality.
What I also like about Genshiken is that it is a good introduction to Japanese college culture as well as "otaku culture."
My second favorite anime is "His and Her Circumstances". It is also a show based on reality. However it is VERY fast paced and somewhat crazy, so it would not be a good place to practice listening comprehension. But it is a really cool show for the first 6-7 episodes.