I have a love/hate relationship with my Canon S3 IS when it comes to shooting action scenes like sports. Well, actually, more like hate-hate-hate-(repeat for 2 years)-like-like-sorta-love.
The S3 IS sports a 12x zoom, which after all the crop factors and sensor sizes are said and done, gives essentially the same maximum magnification as my Canon 20D with a 70-300mm zoom lens. (I've shot identical pictures with both cameras from a tripod to verify.) However, the S3 IS is hampered by its electronic viewfinder and higher noise at high ISO settings, so it's only recently that I've started using it successfully for action scenes. At local pro tennis tournaments, the ushers come down like a ton of bricks on anything that looks like an SLR or big zoom lens, so the S3 with its barely extended lens is perfect for flying under the radar.
That 12x zoom, with such a small light body, makes it very hard to hold the camera steady and track any kind of moving subject. Like I said, it took about 2 years before I was able to track a distant flying bird, or athletes moving around a field. Even just a small turn of the body would put the subject off the screen, and the slow refresh rate of the EVF or the LCD screen meant I would overshoot trying to find it again. With an ultra-zoom camera, you have to tone down your reflexes to move or turn the camera by much smaller amounts than seems natural. Also, it's helpful to develop the technique of zooming out until you can see the subject (even as only a small dot), then zooming in while holding steady on that subject.
For an indoor sporting event, it's a challenge to shoot with available light. I choose from a few techniques here, depending on exactly how much light there is.
As a first step, I ratchet up the ISO setting with the mode dial on "P", and take note of the exposure chosen by the camera when I half-press the shutter. I look for a setting between 1/60-1/125, ideally getting it with an ISO setting of 400 max. If the shutter speed is even faster, like 1/200 or better, I'll reduce the ISO speed as far as possible while keeping in the desired range of shutter speeds. Each step in ISO setting typically changes the shutter speed by 2x, so it's easy to predict what will happen when selecting a higher or lower ISO setting. If the camera is picking 1/50 of a second at ISO 200, chances are it will pick 1/100 of a second at ISO 400.
If there is some combination that produces acceptable results at a decent ISO setting, I'll set the mode dial to "Av" for Aperture priority, then bump the aperture value as low as it will go, typically to 3.5 at minimum zoom, down to maybe 5.6 at maximum zoom. Keeping the aperture value small makes the camera select the fastest shutter speed that it thinks is reasonable, and makes the camera more responsive to the shutter button because it has fewer choices to make when you press the button.
If the camera can't quite do what you want, that is, you can't get a good picture without going to ISO 800, then consider zooming out somewhat until you can lower the aperture setting a bit. The minimum aperture depends on the zoom level, so the camera can get a brighter picture (and faster shutter speed) if you aren't zoomed in all the way. Sometimes, cropping the picture a bit is preferable to a closeup shot that is too grainy or blurry to be useful.
Another consideration is, how close to get to the action. If you are within about 30 metres of the subject, the camera needs a little time trying to focus for each shot. Even if you are farther away, it might try to focus as you lose track of the subject, or something intrudes on the foreground or background. For sports, I find the fastest response when I get far enough away to set the camera on manual focus at infinity, and then just leave it on that setting for all the shots. The same applies to distant wildlife.
Another way to speed up responsiveness is to turn off the "review" setting, which brings up the image for a few seconds after each picture is taken. If you want to snap multiple shots in quick succession (as opposed to continuous shooting where you just hold down the shutter button), you'll have to half-press the button to get rid of the review image, recompose the shot in the viewfinder, then complete the shutter press. Very difficult to do with a moving subject. Set the camera on continuous shooting mode, but also turn off the review feature. Wait for a lull in the action to go back and delete any that didn't turn out.
When composing a sports picture with an ultra-zoom, plan ahead to minimize the amount of panning and tracking, because the EVF/LCD just aren't very good for that. For example, at a tennis match, compose the shot leaving room for the player to move forward as they serve, so that you can hold the camera steady and just press the shutter at the right moment. In the same way, zoom out far enough that you can get a large area of the court in your field of view, rather than swivelling the camera as the player runs back and forth. In a doubles match, you might keep the camera locked on the area up by the net, to capture exciting volleys. With the camera covering a large enough are of the court, you can forget about the EVF and LCD altogether, just watching the action as normal and firing the shutter without looking at the camera at all.
One last tip, this one a bit specialized. The Canon S3 IS has a special "wide-angle" mode that essentially just crops off the top and bottom of the frame, so the picture becomes a panorama, even though the resolution isn't any different than just taking a normal picture and doing the cropping yourself. I've rarely felt the need to use this mode in day-to-day shooting. It's one of the choices for the "Size" button, alongside Small, Medium, and Large.
For sports, surprisingly, I do find some uses for the wide setting. Shooting a tennis match, I noticed every picture had a large area at the top showing nothing but the crowd in the stands. With the panoramic setting, I could focus in on just the court. From a practical point of view, taking a picture of a smaller area makes it easier for the camera to measure the exposure for the important part of the picture, and probably makes the camera more responsive because it's processing and saving fewer pixels. (Although I haven't measured the precise speedup.) Also, if space on the memory card or the computer is an issue, each picture shot this way is smaller than a full-size photo.