St. Luke the Evangelist does not dwell on the journey. He merely states that around the time that the angel left Mary, she set out to the hill country of Judah to visit Elizabeth, who had kept the conception of her child to herself, living in seclusion in the house occupied by her and Zechariah. This story is more than visitation, in the sense that the issues are far greater than a young woman setting aside her own concerns with an unexpected pregnancy to visit her relative, also dealing with an unexpected pregnancy. Both of these women have things in common: they are under a cloud because of their circumstances. Mary is a young, unmarried mother-to-be; Elizabeth is elderly to be considered beyond any possibility of having children. Both of women have been informed by unusual sources of the divine origin of their children.
Mary will sing the song we call the Magnificat, where she proclaims that her soul gives glory to God and her spirit rejoices in the God who saves her. St. Paul says that God chose what is foolish in the world to shame the wise that God chose what is weak to shame the strong. In Mary and Elizabeth we see the best expressions of this.