I've heard that the speed of light is constant, so if an object emitting light is moving, you don't add the object's speed to the speed of the light. The light coming from it will still travel in all directions at the same speed as if the object was still.
But then, how does the red shifting of light occur when an object moves towards us? What is happening to the light in that case? Am i confusing the light's speed with it's wave frequency, and if so what's the difference?