Can expansion of space cause the motion of galaxies, does expansion of space do effect motion of our sun around center and earth around Sun?
No. The solar system is a gravitationally bound system so it is unaffected by the large scale expansion of the universe - see this Wikipedia article for further explanation. Similarly, galaxies like the Milky Way, and even groups of galaxies such as the Local Group, which includes the Milky Way and Andromeda, are gravitationally bound systems and are also unaffected.