Hello. I’m kind of coming into the thread late, but I thought that I’d add my two cents, anyway. I suppose part of the reason that Protestants sometimes accuse Catholics of worshiping Mary is that Protestants and Catholics have different ideas of worship. To Protestants, worship is, essentially, prayer and maybe praise music or reading the Bible. (No offense intended. These are all good things and activities.) Catholics see worship as focused specifically in the Mass, which is centered entirely on Christ and the Eucharist, and in the Eucharistic Adoration of the Body and Blood of Jesus. To Catholics, prayer may be worship if it is addressed to God, but it may simply be a means of communication and reflection if it is directed toward Mary or another saint.
I would also suggest that many Catholics feel like they are praying with Mary or with another saint, rather than to Mary or another saint. Most of the time they are asking Mary or another saint to join them in prayer.
When Gabriel, at God’s command, says “Hail, full of grace, the Lord is with thee” and Elizabeth, filled with the Holy Spirit, says “Blessed art thou among women and blessed is the fruit of thy womb,” both Gabriel and Elizabeth seem to be venerating Mary. Their words form the basis of the most common Marian prayer the “Hail Mary.” When so much of the “Hail Mary” comes directly from Scripture, I don’t think it’s fair to call that veneration of her unbiblical.
It might help, though, if you thought of every Marian devotion and dogma as being about Jesus and not Mary. Mary is Immaculately Conceived, because she is the Ark of the New Covenant (dwelling place of the Word of God) and must be pure and without blemish as the Ark of the Old Covenant was. Mary is perpetually virginal because she is the gate through which God Incarnate entered the world, and no other man is fit to enter through that gate. She is the Mother of God, because Jesus is God Incarnate, Jesus is both God and man, Jesus’ human and divine nature are not able to be separated as they exist in perfect harmony with each other, and women are mothers to people, not to natures. Mary was Assumed because it shows the lifting up to Heaven of body and spirit that Christians hope God, in His mercy and grace, will eventually bestow upon them.
It also might help if you knew that the Catholic Church believes that Mary could only be Immaculately Conceived because Christ’s sacrifice was applied to her before her conception (so she was prevented by God’s grace from falling into the mud puddle of Original Sin that dirties everyone else except Jesus). Mary was preserved from Original Sin by her Son’s merit, not her own, and because Jesus’ sacrifice is what kept Mary clean from dirt in the first place He is as much her Savior as He is ours. You might say that Jesus’ ability to keep Mary without blemish is an example of the kind of cleansing us Christians hope Jesus will be able to achieve for us. In a way, when you understand who Mary was, you see more fully just how special Jesus was, and when you realize the marvels He achieved for her, you are filled with awe for what He can do for you, too.