Grave of the Fireflies is far from whimsical.
I have a 10-movie set of Studio Ghibli films (that's the studio Miyazaki worked for before he retired) and they're all amazing.
Basically, when people like anime and haven't seen the Miyazaki films, I direct them there first and foremost. They're just an amazing starter point (and won't create drooling Narutards.) Plus, I fall asleep prettymuch within the first 20 minutes of any of them because they're so soothing and nice