I watched GITS as a kids and it made such an impact on me that it's forever my number 1 favorite and everything gets compared to it. On a completely different subject, my other tied at number 1 with GITS, is Initial D (original series).
It was Akira that changed me forever, but GITS is still a favorite. If you haven't watched the TV series based on the GITS Manga that came out, they are also really good. Worth owning. Apparently there's a final installment scheduled to come out
this year too!
Some so called "cartoons" I would not recommend in a million years to watch with children like Grave of the Fireflies
This is always the first film I think of when people say "cartoons" since that one is a straight up drama and an excellent story. You get wrapped up in it enough that you forget it is animated because the story is so involving even though the story is so simple. There are newer ones that are in the same vein as well; a more recent one from Miyazaki is
From Up on Poppy Hill - also a drama, and quite good. Not as good as Fireflies, but satisfying. ALL of Miyazaki's films are worth watching though.
I'm watching Sword Art Online (SAO) and am kind of disappointed I didn't check it out sooner. It looked dumb and even the synopsis sounded dumb (it kind of is), but I am fully hooked now. The stories are very good. I'm already 1/3 of the way through after just a few days.