The speed of light is constant, however the frequency of light is variable. Similar thing with sound, high pitched and low pitched frequencies travel at the same speed in the same medium

Red shift and blue shift are exactly the same thing. If you think about light as a wave, like sound, approaching objects have scrunched up waves (blueshifted light) and fleeing objects have spaced out waves (redshifted light). The exact same reason a car horn has higher pitch when it is approaching you and lower pitch when it travels away. The intensity of the effect is correlated to the speed limit of the medium so since light is so fast this effect is typically very small.

Interesting, the interwebs just taught me that redshift does occur like the Doppler effect. Similarity blueshift in the other direction. I always thought it was just from space expanding.

You learn something new everyday!

I'm sorry to have to break this to you, but if you're moving at relativistic speeds, the space between you and objects in your direction of travel does indeed shrink! It's one of the freakier parts of relativity.