The earth goes around the sun. Is the sun moving?

Everything is moving. 

Yes, our sun is also orbiting our galactic barycenter.

Yes. It moves around the centres of gravity with the various planets, and also is orbiting the central black hole of the Milky Way. Meanwhile the Milky Way is moving towards the Great Attractor in Virgo.

It does. Our sun orbits the center of the Milky Way galaxy, and takes about 240 million years to complete one 'galactic year'. Our galaxy likewise moves; we orbit a common center of gravity with about 50 other galaxies called the 'Local Group'; and, the 'Local Group' is part of a galactic cluster as well - which is also moving.

The Sun does move, and with all of the planets in-tow, 

it takes about 230 Million years to make one orbit around 

the center of our Milky Way Galaxy. 

Sure. The Sun is a part of the milky way galaxy and the whole galaxy is rotating so the sun is orbiting the center of the galaxy.

The whole universe is moving. 

Specifically, Our solar system is orbiting the center of mass of our home galaxy.

Yes......……...……………...

The sun also rotates around the common center of gravity for the solar system.

Yes. Even though the Sun is massive, it varies a little due to planet mass.

The effect of Earth on the Sun is a LOT less than Jupiter, 449km, but pic shows the effect of Jupiter.

Plus, our Solar System is orbiting around the Milky Way, and the Milky Way is moving as well. Our Sun's peculiar motion of 368 km/s, and our local group's, of 627 km/s, matches up perfectly with how we understand that all galaxies move through space.  About 2.1 million km/hr