Superzoom Digital Cameras
You're definitely very limited by the $300 budget. There's a reason you see pro sports photographers with big DSLRs and big lenses. That equipment works better when you need speed. It's not just about chnaging lenses, it's also about image quality. Digital SLRs have much larger sensors and even with less resolution their bigger pixels produce better image quality.
But your budget is what it is and we all have to start somewhere. I just wanted you know that you aren't going to get magazine-quality images from a $300 camera. But you can get a compromise that will keep you shooting, albeit with reduced performance. As long as you understand that, you shouldn't be too disappointed.
So what you should look at are superzoom compact digital cameras like the Sony Cybershot H50, the Canon PowerShot S5 IS, the Olympus SP-570, or the Fujifilm Finepix S8100fd. Those cameras all have a lot of zoom, image stabilization, good resolution, and great performance for a compact camera. They will still be pretty poor shooting in the evening or for indoor sports, though. Maybe we can help you imrpove your photos with some better technique, though. No matter what, you'll probably have to adjust your expectations.
Your other alternative is to buy a used digital SLR with a kit lens. You won't have the kind of zoom reach you want but you'll have better image quality and you will have begun the investment in a real performance camera system. Even a digital SLR with the same number of pixels as your current camera will blow it away in low light and in terms of speed. Digital SLRs are just plain better.
I hope that helps. Let us know if you have more questions. Shooting sports is tough. Like I said, there's a reason serious sports photographers invest thousands of dollars in their cameras and lenses. I understand that you don't have that kind of money. Most people don't. But if you're serious about sports photography, making a compromise now for a long term gain might be the better decision :)