Q:

Are red giants stationary or do they move?

A friend and I are having a friendly chat about this. They are stars so I'm claiming they still move in orbit. 

A:

Everything moves.  Entire galaxies move.  Our galaxy is moving towards Andromeda at over 100 km/s and our entire local group is moving towards the great attractor even faster than that.
When a less massive object orbits a more massive one, both move, but the less massive one moves more.  Often the calculation is simplified to make the central body unmoving, but that's a simplification and not accurate.
Our Milkyway has a center of mass with a giant black hole in that region.   If anything in our galaxy didn't move, it would slowly fall into the center.   Stars don't move like that, they move around each other while orbiting within the Milky-Way at high speeds.
Movement, I should add, is defined from a certain point of view, so it's always possible to pick one thing and say "this isn't moving" and that would be correct from that thing's point of view, but that object still sits in spacetime (relatively speaking), and space time defines a force, which (shockingly) moves things, so even if you say "Lets say this object isn't moving", the unevenness of space time will move it.
So, everything moves.   That's a universal law.  

