One thing that has been helpful to me is to list out the passages of Scripture that support Catholic devotion to Mary: Luke 1:28 – “Hail, full of grace, the Lord is with you!”
Luke 1:28 – “Blessed are you among women!”
Luke 1:42 – “Blessed are you among women!”
Luke 1:45 – “Blessed is she who believed!”
Luke 1:48 – “All generations will call me blessed.”
Luke 1:49 – “For the Almighty has done great things for me.”
Luke 1:43 – “And why is this granted me, that the mother of my Lord should come to me?”
Luke 1:46 – “My soul magnifies the Lord, and my spirit rejoices in God my savior!”
Luke 1:48 – “He has regarded the lowly estate of His handmaid.”
Luke 1:30 – “Do not be afraid, Mary, for you have found favor with God.”
Luke 1:38 – “I am the handmaid of the Lord, let it be done unto me according to thy Word.”
Luke 2:35 – “And a sword will pierce your own soul also, that the thoughts of many hearts may be revealed.”
Luke 2:19 – “But Mary kept all these things, pondering them in her heart.”
Luke 2:51 – “And his mother kept all these things in her heart.”
Mark 3:34-35 – “Whoever does the will of God is my brother, and sister, and mother.”
John 2:4 – “Jesus said to her, ‘Dear woman, what is that to me and to thee? My hour has not yet come.’ ”
John 2:5 – “His mother said to the servants, ‘Do whatever he tells you.’ ”
John 19:26-27 – “Woman, behold, your son!” – “Behold, your mother!”
John 19:27 – “And from that hour the disciple took her as his own.”
Zechariah 2:10 – “Sing and rejoice, O daughter of Zion; for lo, I come and I will dwell in the midst of you, says the Lord.”
Zechariah 3:14 – “Sing aloud, O daughter of Zion; shout, O Israel! Rejoice and exult with all your heart, O daughter of Jerusalem!”
Micah 5:2-3 – “Therefore he shall give them up until the time when she who is to give birth has borne.”
Acts 1:14 – “All these with one accord devoted themselves to prayer, together with the women and Mary the mother of Jesus, and with his brothers.”
Galatians 4:4-5 – “But when the fullness of time had come, God sent forth his Son, born of a woman, born under the law, to redeem those who were under the law.”
Ephesians 5:31-32 – “ ‘For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh.’ This mystery is a profound one, and I say that it refers to Christ and the Church.”
Revelation 12:1-6 – “And a great sign appeared in heaven, a woman clothed with the sun, with the moon under her feet, and on her head a crown of twelve stars.” Verse 5: “She brought forth a male child, one who is to rule all the nations with a rod of iron.” From a Biblical perspective, I think venerating Mary is very reasonable. As far as the doctrine of coredemptrix goes, I think there are several passages that support that: Galatians 4:4-5 links the redemption to the birth of Christ from Mary. Luke 2:35 links Mary’s grief to the opening of hearts to let God in. Luke 1:38 indicates that Mary consented before Jesus was incarnated. John 19:25-28 shows that the redemption was not finished until Mary was given as Mother.
Re: Mary’s queenship, there’s quite a bit of biblical support for that too. First, there is Luke 1:28-33, where the angel greets Mary with a royal greeting (“Hail”), she wonders why, and he explains that she will be the mother of a prince (“the Lord God will give to him the throne of his father David”). The idea that she is the mother of a prince supports her queenship, and, since it was an angel who hailed her, it shows that she is queen of angels and queen of heaven.
There’s also an Old Testament prophecy that foresaw a queen standing at the right hand of the Messiah in heaven: Psalms 45:9-15. It even says that faraway nations “will sue your favor with gifts” and that she would be part of a company of virgins in heaven. That supports Mary’s perpetual virginity, btw, and the part about people “suing for her favor” supports prayer to Mary, and the part about her standing at the right hand of the Messiah in heaven supports her Assumption – this is one of my favorite Marian prophecies :p.
Revelation 12:1 supports Mary’s queenship because it shows her in heaven with a crown. In my opinion, the Magnificat song also supports Mary’s queenship, because it includes the words, “He has put down the mighty from their thrones, and exalted those of low degree,” (Luke 1:52), and the song is about Mary.
For all of these reasons (and several more), I think there is plenty of support for the Marian doctrines in Scripture – not to mention for honoring her with special veneration. I don’t think Catholics should be shy about it, either, because we’re really just following the Bible. Anyway I hope that helps. God bless!