I say tackle which camera you want first. The reason being that the particulars of ISO and aperture settings are going to vary depending on what you buy. With a DSLR, PhotoDave has some good suggestions. With say a Powershot though (or any compact point & shoot) many of those tips go out the window.
For example, with a compact, f8 is probably going to be be the smallest aperture available (you won't have f10+). Some shots you'd need to use f8. The sweetspot on a Powershot lens (where it produces the sharpest pics) will actually be more like f4-f5 though. Whether that would be useable for fireworks (or any aperture setting lower than f8) would depend on how close you are (how bright they are) how long you want the shutter open, what part of the show it is (finale is always much brighter), etc.
Also for example, with a point & shoot, ISO800 is going to be really noisy (grainy images). Too noisy to be useable. Long exposures are noisier with a point & shoot too (again much noisier than a DSLR). With a point & shoot, you might actually be inclined to avoid shutter speeds in the 10 second range.
It may sound confusing, but it's really not nearly as complicated as it sounds. It just depends a lot on the particular camera. Once you know what you'll be using, it'll be easier to offer a few good, easy, specific tips (both fireworks, and general flashless night pics).
I wholeheartedly second looking at the Canon Powershots. I use a PowerShot G7 though so I'm totally biased
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The Powershot "A series", "S series", and "G series" cams are great! Great feature sets, nice internal processing of images, great lenses, and a wide range of prices. I'm guessing you're thinking point & shoot?
The one thing I'd say is an absolute must in a compact camera these days is Image Stabilization. Most point & shoots have it now though, so it's not hard to find. A general fireworks tip with IS though is to turn it off when you're using a tripod, or setting the camera on a railing, or a trash can or whatever (not just for fireworks, but any time the camera is already stabilzed on/against something). I know (at least with Canon) that the IS is always trying to stabilize the image. If the camera's sitting on a railing, or a tripod and there's no camera shake to correct, the stabilizer gets fooled into making corrections anyway...slighty blurring your pics. Just have to remeber to turn it off (and then turn it back on of course when you're just holding the camera again).
What are you thinking you want to spend? $100-$300, $300-$500, no limit (don't we all wish
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). Anything you have to have in a camera?...like it has to fit in my shirt pocket, or it has to have 12x zoom, or are you pretty flexible on size/specs?