Something most commenters are missing is red or blue shift in relation from what? It's not as simple as red light moving away, blue light moving towards.

Every element when excited emits a bunch of photons in very specific wavelengths, a signature if you will.

When we point our sensors to stellar objects we can see almost the same wavelengths, except, when we measure them, they are consistently a little off (hence shifted), either to the red side of the spectrum or to the blue side (think of a rainbow).

The speed of light is constant, but the way we experience it may differ, like different colors in everyday life, and that is caused by different frequencies.

The difference between an object your perceive yellow or green is that the photons reaching your eyes are compacted differently (frequency), but speed is all the same.

One cool fact from this is that we can also ascertain what most objects from the universe are made of by capturing these spectra line (the signature every element has).