First off, where are you? I don't know if you are looking to spend $300 or 300£, and not knowing the conversion rate, I can't tell you how they compare either.
Blurring is usually caused either by subject motion or by camera motion. It's usually easy to tell by looking at the photos, but the easiest remedy that will help with both is a faster shutter speed, which often means using a higher ISO setting.
There are several cameras that should do a decent job for you. Right around, or even under, 300USD you can get one of the Canon ELPH models. These cameras are well built and have great image quality (for a pocket camera). Canon incorporates a Kids and Pets mode on their cameras, which is intended to help with faster moving subjects.
There is still an issue of shutter delay, although not nearly as bad as it was a few years ago. I remember some cameras that would take over a full second to take the shot after the shutter button had been pressed. The lower priced cameras usually have more delay, and the more you spend the better it is for you.
Depending on your level of experience, or your sense of adventure, you may also consider looking for a used SLR. You will eliminate virtually all shutter delay, and you will have they opportunity to set your exposures precisely as you see fit. These cameras can be intimidating at first, but they can be used in a variety of auto or semi-auto modes depending on your comfort level. They aren't pocket cameras, but the potential in terms of image quality, and more, is far above the point and shoot models.
I hope this helps a bit, and again, welcome to the Zo.
There are two types of photographers, those that shoot Canon & those that wish they could.