Is the sun moving?

Yes. The Sun spins on its axis just like the Earth. It (and the rest of our solar system) also revolves within the Orion arm of the Milky Way Galaxy. The Milky Way Galaxy is also moving through space, along with many other galaxies in the so-called Local Group. The motion of all these galaxies is generally toward an unseen concentration of mass, in the direction of the Norma Galactic Cluster.

The Sun's surface also ripples and pulsates, both cyclically (p-waves) and gravitically (g-waves), likely due to churning and the different rotational velocity of its core. The major surface motion is the development of solar flares initiated by magnetic flux.

It takes about 225 million years or one cosmic year for the sun and nine to travel around the milky way once.