The Doppler Shift – how we use it
Atomic energy transitions leave features in the spectrum whose rest wavelengths are known from laboratory work. We can measure observed shifts in these wavelengths from astronomical objects, and see how fast they are moving (you only get the line-of-sight motion: towards or away from you).
More subtle analysis can also yield other motions, like rotation or turbulent motions. These are all direct uses of the Doppler shift. It doesn’t matter how far away the source is, either.