I'm curious, does the Sun actually move? The concept behind the question is, if you wear a shirt and you move, does the shirt actually move? Like the Solar system orbits around the Milky Way, but does the Sun move?
This is to do with Frames of Reference. The video below gets across nicely that all motion is relative, your perception of it depends on your point of view:
Frames of Reference - Hume and Ivey, 1960, Educational Services Inc.
If we chose the Sun as the center point of the Universe, we could say everything moves around it - but that doesn't work so well because the movement of stars in our galaxy is much more easily understood if the galaxy's center of gravity is used as the fixed point. Also, stars are massive enough that the ones near each other influence each other's movement a lot and they tend to sort of weave around one another - the study of this is called stellar dynamics.
Equally, the movement of planets in the Solar system was extremely confusing until we accepted the Sun as the center of motion, and saw it as the local fixed point, for the Solar system. (Actually that isn't entirely accurate either - the center of gravity of the Solar system is the point everything moves around, it's common barycenter - see this question.)