Professional sports photographers have big zoom lenses and other pieces of fancy equipment that allow them to get close-up shots all over the field when they are shooting a game. You may not have all that, but there are still a few things you can do to get good photos at sporting events. The big thing in sports photography is to anticipate what is going to happen, and to be prepared. When you are shooting action, there is no time to do anything other than push the shutter button.
It is good to have some level of zoom lens if you are going to try to shoot sports. Most point-and-shoot cameras have zoom lenses, and Digital Single Lens Reflex cameras have zoom lenses available. You may not have what the pros use, but use what you have. With some care and practice, you can get good pictures of sporting events even with a point-and-shoot camera. Consider a mono-pod to add stability to your shots.
Most cameras today have auto focus, and some will allow you to keep a moving object or person in focus. If you do not have a top-of-the-line digital camera, you may have trouble because most of them won't shoot if they are still trying to focus. When shooting sports there is no time to focus. A trick here is to focus on an area, and wait until your object moves into that area to shoot. You can also focus on one person and follow that person throughout the play, or until you take the picture.
Timing is everything in sports photography. Even half a second can be the difference between a great photo and something out of focus. Try to anticipate where the action will go. This takes practice, and even the pros don't always guess right, but you can train yourself to anticipate where the action will reach its peak. If you are taking pictures at a game, it is a good idea to watch the first several minutes without shooting, just to get a feel for how people are moving around. Understanding the sport and how movement happens is also important. The more you know about the sport the more you can anticipate action.
Most cameras have a "burst" mode, which means they shoot several pictures in a very short amount of time. Set your camera on the highest setting so it will fire rapidly when you are shooting the action. It's better to have 30 pictures almost the same than to have only one and it not be in focus. Set your camera on the highest shutter speed lighting conditions will allow to stop action. When shooting action, you need to stop the action, so the higher the shutter speed the better off you will be.
Try something different
Most people stand on the sidelines, and that is where you usually do get the best photos. But try going other places, such as the end of the court or field, or up high into the stands. Often at games all the photographers are standing in the same place, and they all get the same pictures. If you want something different, you will have to do something different.