If you haven't used the manual settings on your camera, you may need the manual to help you adjust a few features that will help you with your night time shots.
Understand how to adjust your camera's ISO settings. ISO is the digital equivalent to film speed, so the higher the number, the less light you need to shoot sharp pictures. The downside is that higher ISO settings also produce more grain (or digital noise), which is especially apparent in dark areas of an image.
Use your camera's lowest ISO setting, which for many models is 100. Lowering the ISO means your camera's sensor is less sensitive to light, so you'll need longer shutter speeds to achieve the proper exposure. With regard to fireworks pictures, this means you'll see some trails of light as the fireworks move through the sky, and with some practice you'll learn to use this property for pleasing effects.
Controlling your shutter speed is also important. Most point-and-shoot cameras have long exposure modes that let you shoot one-second (and much longer) exposures. Start your experiments with a one-second exposure, review your pictures, and adjust the shutter speed as necessary.