Did Mary and Joseph went to Bethlehem for the birth of Jesus because they were following the stars?

No, Mary and Joseph went to Bethlehem because they were required to register for a census ordered by the Roman Empire. The journey was not motivated by following stars; rather, it was a legal obligation. The Star of Bethlehem is a symbol associated with the birth of Jesus and the visit of the Magi, who interpreted it as a sign of his birth.