One common misconception of Catholics is that we worship Mary. We do NOT worship Mary. However, we DO honor Mary. She is God's greatest creation. Remember, only God and Mary can claim Jesus as their son. Here's some verses to explain why we honor Mary as we do.
As early in the Bible as Genesis Mary is prefigured: "I will put enmity between you and the woman, and between your offspring and hers; He will strike at your head, while you strike at his heel." Genesis 3:15
God speaks these words to the serpent in the Garden, showing that a woman would come that would bring forth a child to strike at the head of the devil.
"When the wine ran short, the mother of Jesus said to him, "They have no wine." And Jesus said to her, "Woman, how does your concern affect me? My hour has not yet come." His mother said to the servers, "Do whatever he tells you." John 2:3-5
Jesus addresses his mother as "Woman", not out of disrespect, but to show that she is the "woman" of Genesis 3:15. We see that the ministry of Jesus begins at the request of Mary. She gently asks her Son, and he does not say no. Mary is always telling us to do whatever Christ asks of us.
She is predicted as mother of the Messiah in the Old Testament: "The virgin shall be with child, and bear a son, and shall name him Immanuel." Isaiah 7:14
The first chapter of the Gospel of Luke talks about the Annunciation and the Visitation. "Hail, full of grace! The Lord is with you." Luke 1:28
Notice how the archangel Gabriel's greeting is the beginning of the Hail Mary. Our belief that Mary was without sin, original or actual, is supported by the angel calling her "full of grace". Sin is the loss of grace; one who is full of grace has no sin.
"Do not be afraid, Mary, for you have found favor with God." Luke 1:30
"Behold, I am the handmaid of the Lord. May it be done to me according to your word." Luke 1:38
This is what Elizabeth says when Mary comes to visit her: "Most blessed are you among women, and blessed is the fruit of your womb. And how does this happen to me, that the mother of my Lord should come to me? For at the moment the sound of your greeting reached my ears, the infant in my womb leaped for joy. Blessed are you who believed that what was spoken to you by the Lord would be fulfilled." Luke 1:42-25
Note that another part of the Hail Mary is in Elizabeth's greeting. Also note the term "How does this happen that the mother of my Lord should come to me?" This echoes King David when he says, "How can the Ark of the Lord come to me?" (2 Samuel 6:9) More on that later.
"From this day all generations shall call me blessed." Luke 1:48
Simeon predicts the sorrow she will feel at her Son's crucifixion: "You yourself a sword shall pierce." Luke 2:35
Mary was not pierced by an actual sword, but a sword of sorrow in her heart.
"When Jesus saw his mother and the disciple there whom he loved, he said to his mother, 'Woman, behold your son'. Then he said to the disciple, 'Behold, your mother.'" John 19:26-27
Jesus not only gives his mother to John so that he can take care of her, but to the whole church, that we might take her as our mother.
Mary is the Queen of Heaven, Mother of the Church, Mother of our Lord. She is the greatest of God's creatures. She is:
"A woman clothed with the sun, with the moon under her feet, and on her head a crown of twelve stars." Revelation 12:1
This calls to mind the image of Guadalupe. But what does it say right before that?
"Then God's temple in Heaven opened and in the temple could be seen the ark of his covenant. There were flashes of lightning and peals of thunder, an earthquake, and a violent hailstorm." Revelation 11:19 It goes right from talking about the Ark of the Covenant to Mary. This would be incredibly important to the Israelites, because the ark contained the Word of God (the commandments), the manna (bread from heaven) and Aaron's rod (the symbol of priesthood). Why does it go from talking about that into talking about Mary? Because Mary is the new ark of the covenant. She held within her the Word of God, the Bread of Life, the High Priest. How do Mary and the Saints intercede for us? Where is that in the Bible?
"Each of the elders held a harp and gold bowls filled with incense, which are the prayers of the holy ones." Revelation 5:8
"Another angel came and stood at the altar, holding a gold censer. He was given a great quantity of incense to offer, along with the prayers of all the holy ones, on the gold altar that was before the throne. The smoke of the incense along with the prayers of the holy ones went up before God from the hand of the angel." Revelation 8:3-4
We can see from these verses that the angels and saints are interceding for us, offering our prayers to God. And from all the miracles, too many to mention, we can see that the Lord is still working through them.