Possibly, your best choice could be a used, older model of DSLR like a Canon 40D, for example. I see prices from about $250-$300 for the body, then you have a some money to find a used zoom lens. It will not be the best and easiest equipment for some situations, but it is a good start, and you can build an improve your set of lenses over time.
Using bounce flash or flash on a light stand can be a helpful way to make moving poses clear. Otherwise you need to shoot in plenty of light to get a good shutter speed 1/250 or faster. Most point and shoot cameras stay in auto focus, so there will always be a delay when you hit the trigger, waiting for it to focus. DSLR cameras can have the auto focus turned off after you find the focus ahead of time, for instant shutter release action.