your friend says, "since light is a wave the color can look blue or red depending on whether something is moving towards or away from us. A red star therefore has to be moving away from us". How would you respond? Stars that are red are that color because of the surface temperature AND the fact that they are moving away from us.

Light is indeed a wave, but a red star is that color because of the surface temperature only. Correct, light is indeed a wave and when something is moving away it looks red.