In case anyone is interested, Soderbergh has posted on his Extension765 website a black and white and mute version of Raiders of the Lost Ark, as a way to make people notice how well staged the movie is. Oddly enough, it looks amazing, with that high-contrast lighting.
It's a great workout to try and keep an eye out for staging.

I find that true for some movies, but not others. In experimental movies, the image and sound themselves, along with the way they entwine, are themselves the subjects of the movie. And a lot of art movies have no problem whatsoever in putting the cinematography in the way of the narrative, making it as much a focal point for the audience as the story itself sometimes, as long as they feel the movie is better for it.
For most comercial movies, though, that quote seems to me to be correct: cinematography is there to help tell the story in the most unobtrusive and effective way possible.

I agree with Guy and Nick. I own the G7 as well, and the 14-42 kit zoom is very good. It's a short range, and it's very slow (f/3.5-5.6), but for handheld work in daylight it works awesome - only maybe a tad *too* sharp, at least for my taste. There's also the 14-140 Lumix lens, but I can't speak about it, as I've never tried it. The 12-35mm f/2.8 Guy is talking about is excellent, but costs as much as the camera itself. If you look carefully, you might find some good deals in old used manual zoom lenses - won't give you such a wide angle, but will definitely be a lot cheaper.

Beyond that, I'd say a good place to start would be a a fast 50mm, and a fast wide-angle. I find that telephoto lenses are not as important in a micro four thirds camera, as a 50mm gives you around 110mm equivalent focal length anyway. But that will depend on what you plan to shoot, of course.

As Nick said, give preference to APS-C or full-frame lenses, as you can always pick up a speedbooster or get a new camera. The advantages you get from Panasonic G lenses are auto-focus, remote operation and stabilization - you be the judge of what is the compromise you're willing to make.

For fun, you can also look into the crazy world of C-mount lenses, where you can find some flawed little gems for super cheap. The Fujian 25mm f/1.4 and the Fujian 35mm f/1.7 are easy to find, very cheap and super fun. No good for professional use, though.

Completely disagree with that definition, to be honest. Everyone has seen boring beautiful movies: pretty cinematography doesn't make a movie automatically enjoyable. And everyone has seen "ugly" masterpieces, as in "not technically perfect or well lit or well shot, or with good composition" and all that (late Godard, 80's Jarman, etc.).

"Good" cinematography is different from "technically perfect" cinematography, and different from "beautiful" cinematography. Great cinematography complements the directing and the script, so it can stop being great if the movie is altered.

A better way to explain "cinematography" is that it encompasses all aspects related to the creation of the image: camera placement (close-up, medium-shot, wide-shot, aerial, whatever) camera choice, lens choice, choice of time of day to shoot, choice of colors, choice of lighting, help decide where to place the actors so they look good, camera movement, and so on.

You should also add the Sony a6000 to your list of options. Looks awesome, and now that the even better Sony a6300 came out, the a6000 may get a bit cheaper (and it's already a bargain, for how good it is).

I own and love the Panasonic G7, and it's the first decent camera I've ever had, so maybe I'm biased. Anyway, I think it's the perfect starting camera, as it is very versatile: light, good for video and photography, you can carry it around everywhere and all the time. It is un-complicated if you want it to be, but it's also able to produce a great looking image if you know your way around it.
Around your budget, there's the BMPC, but the crappy sound, horrible battery life and the huge file sizes makes it too much of a "only-for-short-movies" camera for my taste, if you know what I mean (then again, that might be exactly what you're looking for).
The Canon cameras at that price all feel way too soft to my eyes (not necessarily a bad thing, that's where taste comes into play), even though they produce good color very easily.
Within your budget, there's plenty of good picks in the Panasonic brand, like the GH3, or the GH2, or the G7.
That being said, it's all a matter of what you want to do. The G7 is small but packs a punch with a very sharp 4K image, and the battery life is crazy good. But it has no headphone output, and it's not weather-proof (the GH3 is both). Both cameras would require an external recorder or pre-amp if you want good audio out of them, so maybe they are not what you are looking for. Some cameras have built-in XLR inputs, though, and you might find them used within your budget, if you take the time to look around.

It's really all about what you would be using it for. If you want a versatile and easy camera with great video, then I agree with Guy that the G7 is currently the best way to go. If you're looking for a camera with even greater video capacities (except 4K), there's the BMPC. For an all-around workhorse, maybe look for a camera with XLR inputs, internal ND filters and a good range zoom lens (which would have to be used, to fit in your budget).