What a great question! The sun is actually rotating just like the earth does but takes about 30 days to do a full rotation (compared to on earth where 1 rotation is equal to 1 day). Also on top of that, our solar system (the sun and all the planets) are in an orbit in the much bigger Milky Way Galaxy which holds many stars. So yes, the sun is moving and is not stationary in space.

Hope that helps!
Thank you,