We know that universe is expanding and galaxies are moving away from each other. Does this mean the galaxies are also expanding in itself and therefore I guess growing larger in volume? Depending on the answer of that, does the solar system also expands?
Metric expansion works (for now) on large scales only. This is because matter becomes clumped together by gravitation, which fights the expansion. These clumps of matter may continue expanding more slowly, or they may begin to contract, depending on the balance of gravity to cosmological expansion.
The gravitational interactions inside the Local Group of galaxies are large enough that there is no measurable cosmological expansion going on. For example, the Andromeda galaxy is within gravitational distance of the Milky Way, so the two galaxies are actually falling together, not expanding away. Further out, between galaxy clusters, is where the metric expansion is really significant.
So, no--the galaxies and solar systems do not follow the metric expansion. (Not that they couldn't expand for some other reason.)
See http://en.wikipedia.org/wiki/Metric_expansion_of_space for a lot more info.