The clearest picture of praying to Mary is the wedding at Cana. The wine runs out. She notices before anyone else, and she does two things:
And the wine failing, the mother of Jesus saith to him: They have no wine. And Jesus saith to her: Woman, what is that to me and to thee? my hour is not yet come. His mother saith to the waiters: Whatsoever he shall say to you, do ye.
John 2:3–5, Douay-Rheims
She tells her Son what is needed, and she tells everyone else to do whatever he says. That is the entire shape of it. She asks Jesus, and she points at Jesus. Every prayer to Mary is a way of standing where the waiters stood.
Catholics honor Mary; worship belongs to God alone, and no Catholic prayer to her asks for anything she could give on her own. The Hail Mary is the plainest case. Its first half is Scripture, the angel's greeting and Elizabeth's, word for word. Its second half is one request: pray for us sinners, now and at the hour of our death. Pray for us. The same thing you ask of a friend.
The honor itself is in Scripture, in her own mouth:
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.
Luke 1:46–48, Douay-Rheims
Calling her blessed is doing what that verse says every generation would do. The Church's own words on her place, and on why devotion to her leads toward her Son rather than away from him, are in the Catechism, paragraph 971, and paragraphs 2673 to 2679.
The Hail Mary is the one to know first. The Memorare is for when you need to be reminded that asking her has never been useless. The Hail, Holy Queen is what the Church has said to her at the end of the day for eight hundred years, and it has three sisters for the other seasons. The Rosary is a walk through the life of Jesus with her beside you, one Hail Mary at a time.
The site keeps an index of the times she is reported to have appeared, with what was said and what was going on there at the time. What she asks, when she asks anything, is prayer and a turning back, and she names her Son. It is at Heaven's Voice.