Remember, if you use MANUAL focus, the shot is pretty much instant.

I have a FujiFilm FinePix 6900Z -- over double what you want to spend -- but I just bought my father-in-law a Canon S330 for Christmas on ebay for $300.

However, for sports photography, you may want to have the ability to use an external flash.

Just like suggested above, visit DPreview.com (Buying Guide->Features Search) to select all the options you want to see what cameras fit your needs. You may need to shell out more than $300 unless you can find a nice used one (on ebay for example).