What do I need to look for to take good action photo's?

My daughter is in gymnastics and my current camera misses the action, or more often, blurs it. My son plays basketball and I have the same problem. What do I need to look for in a camera that will take good action/moving photo's? I of course would like a decent zoom as well. HELP!

My son is in gymnastics too and the gymnasiums usually have low light. Also, they don't allow to use flash during the competitions and usually we can not get close anyway to use the flash. So the only way to get satisfactory pictures is to use a digital SLR and very high speed interchangeable lenses. For example: Nikon D50 and a couple of fixed lenses. Like Nikkor 1.8 50mm automatic and Braun 2.8 135mm manual lens. Those lenses are available for around $100 a piece. There are zoomable high speed lenses available too, but most of people can not afford to buy them.