Good question! Yes, when we look at the spectrum of most galaxies, it's true that the lines appear to be red shifted. This led to the whole concept that if almost everything is going away from us, then that is pretty good evidence for the universe expanding.
However, if you look more carefully at a spectrum of a galaxy, especially if it's seen partly edge-on, you can tell about rotation. This is done by taking a spectrum of the light emitted from let's say the LEFT side of the galaxy. You'd still see the overall redshift caused by the entire galaxy moving away from you, but you could for example see that it is slightly MORE redshifted. That would mean that on the left side, it's going away from you a bit more. Then if you take a spectrum of the right side of the galaxy, you see that it's redshifted less. That means you can infer that it's coming towards you a bit more. From this we can tell about the rotation speed of a galaxy. You can do this only for light being emitted, though, which means you miss out on any of the stuff that's NOT giving off light.
That leads to evidence for dark matter existing, because the rotation properties of the visible stuff we can see (stars and gas) show that they don't have nearly enough mass to keep the galaxy rotating the way it is. There must be something else there with mass that's keeping it together - thus leading to dark matter.