The Sun, and the entire solar system, is orbiting around the center of our galaxy. There's nothing special about the Sun, and it is not at a fixed point, it "circles" around the center of our galaxy, and the entire galaxy is moving too, relative to other galaxies, etc. So far it seems to be a
Great Attractor (like a gravitational "centre" when looking at the entire observable universe), but that's a recent observation, still in research.
In a computer animation, one can pick any point where to put the camera, and that point can take the Sun as "fixed". Funny thing, when you attach a camera to something that moves, the video might look very unusual for us, who are used with a "fixed ground" as a reference. See for example this 3 seconds animated image with a camera attached to the track of a tractor. Play it in a loop.
https://www.eevblog.com/forum/chat/fun-for-nerds/?action=dlattach;attach=815313Look at the yellow wheel(s). Notice how the yellow wheel normally rotates, then the rotation "instantly stops", as if it where no inertia!?!
In that video loop there is no mystery of how's so, that a rotating wheel can (apparently) "instantly stop". Only bringing it to illustrate the weirdness that can arise from the movement of two reference frames relative to each other.
As for the objects expanding together with the universe, my understanding is that the inflation is happening mostly for the empty/interstellar space, not for planets and stars and other objects. So, the computer, the Earth, us humans, do not "expand" with time.
Keep in mind that I'm not a physicist, so I might have got some aspects wrong.