Anticipate action by trying to predict where the subject/action will go, and then be ready to shoot it when it moves into the frame of your shot. Think ahead and get positioned for the action that's to come.
Let action happen within the frame. Don't constantly move the camera in an futile attempt to catch everything.
And don't be afraid to allow your subject to move out of frame, rather than trying to follow them with your camera.
This is especially important if you're taking a shot of a person who is walking and then later another shot of the person sitting down.
If you follow the person while they walk with your first shot and always keep them in frame, and then cut to second shot of the person sitting down, it can create a mental disconnect for the viewer as to how the person got to the second position.
If instead you show them walking out of the frame in the first shot, then it's logical to the viewer that the person would be seen in the next shot sitting somewhere else.