[Arvon]

In most cases of the visible stars you wouldn't notice the Doppler shift, sometimes known as the red shift. The relative speed is not enough to see it. You would need spectrographic equipment to note it in the visible stars. Remember that other than Andromeda all of the stars you can see are stars of the local system. Yes you people in the southern hemisphere can see the Magellian clouds.

By the way the fastest star one can see is Barnards Star, which is a Brown Dwarf.