The key to taking good shots of fast moving subjects is understanding the way your camera works. the three main variables invoved when taking a shot are ISO setting, SHUTTER SPEED and APERTURE SIZE. Most good cameras shouls allow control of all three.
The ISO setting controls the camera's light sensitivity. the higher the setting the more sensitive to light the cameras sensor is, although images become a little noisy. Always use a minimum ISO setting of 400 for action photography, in cloudy conditions you might raise this to 800 and at night even as far as 1600 if you camera allows it.
The faster the shutter speed the better the camera will be able to take photos of fast moving subjects. The catch is that a fast shutter speed also allows less light to enter the lens, which in darker conditions (ie at night games) may not be sufficient to produce a decent image.
The aperture size (often referred to as the f stop number) controls the size of the aperture that lets the light ont othe cameras sensor. The larger the aperture the more light comes in, allowing a faster shutter speed. However a large aperture (low f-stop number) also reduces the depth of field, meaning that objects in front of or behind the object that you are focussing on won't appear as clearly.
For sports photography because you are trying to capture fast moving subjects you genrally want to use the highest possible shutter speed. For this reason the main feature to look for in your camera is a fast lens (low minimum f-stop setting) most cameras will have this printed on the lens (it will say something like "f 2.8 - 5.6", the first number is the important one. 2.8 or lower should give decent results. Note that small compact cameras genrally have pretty small lenses meaning that their maximum aperture size isn't all that large, so they are often not as good as larger cameras for shooting fast moving subjects. They are also less likely to offer as much manual control of camera settings.
To ensure your camera uses the fastest practical shutter speed you need to set the camera to APERTURE PRIORITY mode, this allows you to manually choose the aperture setting and the camera will choose the appropriate shutter speed based on the light levels. Set the aperture as low as possible (say 2.8 or lower) and the camera will automatcally pick the fastest appropriate shutter speed. Many cameras have a preset action photography mode which works in this way and provides good results.
This should do the trick in daylight, however at night the light levels may not be sufficient causing the camera to choose too low a shutter speed. If this is the case you may have to set the camera to SHUTTER PRIORITY mode, and manually experiment with forcing a fast shutter speed. This will likely result in the photo being a bit dark, but you can fix this up in your camera software to a point.
Hope all that made sense and is helpful, any other questions feel free to ask...
