Fireworks with showers and trails of fine sparks can be taken with a hand held camera provided that the lens is fast enough. The more you zoom, the slower the lens becomes.
Fireworks with isolated shooting stars are best taken with the camera on a tripod. A long exposure gives long trails; a short exposure gives short trails.
I haven't memorized the needed lens openings for manual cameras. Automatic cameras are tricky because they react to the amount of light in the scene particulary near the center. If there are lots of fireworks bursting, you may get a picture that is darker overall but often with better color.
A "fireworks" setting is not necessarily the best for other kinds of night photographs.
But usually you will want some kind of "night" setting for night shots. This also depends on whether you want to take ordinary pictures in low light (often needs a higher ISO) versus take pictures of nighttime scenes (this is what night mode is for).
If you don't have a night mode and your night shots are "too bright" and washed out, set the exposure compensation (intentional lightening or darkening) to minus two. Experiment with minus three (if the camera has that) or minus one if the picture got too dark.
For cameras with spot metering or center weighted metering and appropriate features, if you aim at a brighter part of the scene, press the button halfway, then frame your shot and shoot, you get a picture that is darker overall. Or brighten up the picture if you first aim at a darker part of the scene. You can either use this technique in lieu of exposure compensation or to obtain additional exposure compensation, or accidentally use this technique and get unpredictable results with or without exposure compensation.
Digital camera hints:
http://members.aol.com/ajaynejr/digicam.htm