I'm curious; does the Sun actually move? The concept behind the question is that if you wear a shirt and you move, does the shirt actually move? The Solar System orbits around the Milky Way, but does the Sun move?
The Sun orbits in the Galactic potential. The motion is quite complex; it takes about 230 million years to make a circuit (meaning an orbital speed of around 220 km/s), but at the same time it oscillates up and down with respect to the Galactic plane every $\sim 70$ million years and also wobbles in and out every $\sim 150$ million years (this called epicyclic motion).
One can also measure how the Sun moves with respect to the average motion of the stars in the solar vicinity - the so-called local standard of rest. Estimates of the Sun's motion with respect to the LSR vary a little. According to Dehnen (Walter!) & Binney 1998 the Sun moves at 10 km/s inwards, at about 5 km/s faster than the average star tangentially to the Galactic centre and at about 7 km/s upwards out of the Galactic plane. Though as I mentioned previously, these motions are of an oscillatory nature around the mean value.
We could cast our net wider and ask how fast Sun moves with respect to the standard of rest defined by the cosmic microwave background - the local cosmological rest frame. The Sun moves at about 370 km/s relative to the CMB in the direction of the constellation Leo (as revealed by the dipole-like anisotropy in the CMB temperatures).
Yes. No. It depends on your perspective. Suppose you're in a plane, flying from New York to Los Angeles. If you stay in your seat all the way, you haven't moved (from your seat). On the other hand, you clearly have moved, since you were in New York and now you're in Los Angeles. Motion is relative to some reference point, and you can always pick a reference point that makes something stationary. However, some reference points are easier to use than others; that's why the geocentric model of the universe isn't used any more: cycles and epicycles and so on are just too complicated, and using the sun or the the center of the galaxy makes it much easier to visualize and calculate what's going on.