The light travels at the same speed, regardless of the motion of the object emitting it, but the wave-fronts of the light are either pushed closer together (resulting in a shorter wave-length) or pulled further apart (resulting in a longer wave-length), shifting the light colour towards blue or red.
Light travels fastest in a vaccuum, and slightly slower in denser materials, such as air or water. Different wavelengths travel at slightly different speeds in dense materials too, which is why the component colours of light appear to split in a glass prism.
