Sichuan and Hunan foods are more different than one might imagine.
Sichuan is the home of "strange-flavor," "numb-spicy," and "fish-scented" dishes--
guai wei, mala, & yuxiang respectively. You can find these dishes at all sorts of places in Chinatown and across China, not just in Sichuan. Ed's Potsticker House has gotten plenty of love for its yuxiang eggplant, despite being a Northern-focused place. "Guai wei" is (sometimes) typified by bon-bon chicken which you see around town. At some places, it means doused in chili oil, but this isn't quite correct. I've enjoyed it at Spring World (though I confess, I can't recall what they call it in English

).
Hunan I'm less familiar with, but still had a few wonderful experiences with it. A common feature, contra Sichuan's dried chiles, is a preference for fresher, whole/chopped chiles, or the salted ones, called
duo jiao or
duo la jiao. Lao Hunan does a quintessential whole-fish dish covered in these chopped salty peppers. The emphasis is more on the salt than the heat, and it was quite good when I had it--authentic, at least, if a bit salty for some tastes. I think it's called Hunan-style whole fish on their menu, but don't quite me on it. Lao Hunan also features
chou doufu, or stinky tofu, another dish represented all over China but for which Hunan has some renown. I couldn't stand it over there, and wouldn't order it here, but the adventurous might be intrigued. I'm just impressed they offer it.
I haven't been to Lao Beijing in a while to comment on their more Northern dishes. The classical emphasis is, as CracyC mentions, more wheat-based. Two such dishes (from Ed's Potsticker House, at least) that typify the cuisine to me are
zha jiang mian & da la pi. The former is probably familiar. The latter is a pile of wheat gluten noodles, I believe, dressed with some pork, and maybe some vinegar and sesame oil. Typical Beijing/Northern/Northeastern flavors are garlic and green onions. I'm still looking for adequate
baozi to rival what I ate over there, and despite its lesser reputation, miss Beijing food quite a bit.
Can't help too much with Shanghainese food, sadly.