Do Stars Move In The Sky?

Yes, all stars move.
The stars also have their own proper motion through space.
Our Sun moves at a speed of 19 kilometers per second through the Milky Way.
In addition the Milky Way, like all other spiral galaxies is rotating.
Also the stars move on orbits around the center of their Galaxy.
This spinning motion of the galaxy gives the Sun and the stars near it an orbital speed of about 250 km/sec.