Hi,
Welcome aboard.
Sports photograph needs a longish lens on the camera, and a fast lens as you'll need high shutter speeds to stop the action. You won't get them in elderly compact cameras. The older ones are mechanical and the shutter press has to do a lot of work, and that means it takes its time. So you'll not get the shot you want.
Mostly because of those factors, people go for a long, fast lens on a SLR. That, I'm afraid, will be nothing like compact and expensive. But look around and you'll see lots of decent SLR's that are not as dear as you'd expect*. Get one with shutter priority and a medium fast and medium length lens (say 135 or 200mm) and you ought to be OK for sports photography. It depends mostly on how close you want to be to the action and how much you have to spend.
BTW, at sporting events compacts can be useful, you use them instead of changing the SLRs lens and wasting time on it. A good rule at sports events is that with two lenses the wrong one will be on the camera when you really need it. So two cameras work well together.
Regards, David
* Have a look at the Minolta 7000, for example; although it may not be called that where you live.