And Mary rising up in those days, went into the hill country with haste into a city of Juda. And she entered into the house of Zachary, and saluted Elizabeth.
And it came to pass, that when Elizabeth heard the salutation of Mary, the infant leaped in her womb. And Elizabeth was filled with the Holy Ghost: And she cried out with a loud voice, and said: Blessed art thou among women, and blessed is the fruit of thy womb. And whence is this to me, that the mother of my Lord should come to me? For behold as soon as the voice of thy salutation sounded in my ears, the infant in my womb leaped for joy. And blessed art thou that hast believed, because those things shall be accomplished that were spoken to thee by the Lord.
And Mary said: My soul doth magnify the Lord. And my spirit hath rejoiced in God my Saviour. Because he hath regarded the humility of his handmaid; for behold from henceforth all generations shall call me blessed. Because he that is mighty, hath done great things to me; and holy is his name. And his mercy is from generation unto generations, to them that fear him.
He hath shewed might in his arm: he hath scattered the proud in the conceit of their heart. He hath put down the mighty from their seat, and hath exalted the humble. He hath filled the hungry with good things; and the rich he hath sent empty away. He hath received Israel his servant, being mindful of his mercy: As he spoke to our fathers, to Abraham and to his seed for ever. And Mary abode with her about three months; and she returned to her own house. (Luke 1, 39-56)
If I haven't fallen asleep by now, I have a fairly easy time imagining this Mystery. Mary, having been told by the Archangel Gabriel about the pregnancy of her elderly kinswoman, immediately gets up and sets out to her house. She knows Elizabeth will need assistance in the last few months. The Bible doesn't say if Joseph went with her or not; I always imagine her alone on the journey. What was she thinking as she traveled? Practical things perhaps; or maybe wondering how she was going to tell Elizabeth about the Angel. No doubt she prayed too. And then she sees Elizabeth and calls out to her. That's when all the interesting things happen: Elizabeth's child (St. John the Baptist) leaps in her womb and being filled with the Holy Spirit calls Our Lady the "Mother of my Lord" and praises her for believing. Mary in turn praises God in her beautiful Magnificat for all the wonderful things He has done for her (and us). Her prophecy that the world shall call her Blessed has indeed come spectacularly true.
Fisheaters.com has a great page about the Visitation. I particularly like the parallel between Luke and II Kings; that Mary is the Ark of the New Covenant.
The fruit of this mystery is charity towards our neighbor.