The biggest problem with most point-and-shoot cameras, when you're wanting to shoot action, is shutter lag. What that means is there is a momentary lag between the time you push the shutter release and when the picture is actually taken. It may be just a fraction of a second, but that can be significant when you're shooting pictures of something that's moving as fast as our dogs sometime do. What you end up with is lots of pictures of an empty space where something interesting was happening when you pushed the shutter, but was gone when the shutter actually opened.
I studied photography at Brooks in Santa Barbara (in the olden times when people used film,) did photography professionally for a time, then taught classes and sold cameras. I sucked as a camera salesman because I would talk people out of buying a bunch of lenses before they even knew what they needed.
The lens is more important than the camera itself, so get a good, short zoom - maybe in the 35-105 mm range - and get really comfortable with it before you even think about getting another lens.