Matthew was actually the second gospel written, around 10-20 years after Mark, and all the gospels were written after Paul's letters. So like the Old Testament, the New is also not in chronological order. The order in both cases seems to have been chosen for literary and theological significance, rather than chronology. Matthew's Gospel appears first to introduce us to the (paternal) genealogy of Jesus.¹ What at first glance looks like another long, dull list of begats turns out on closer inspection to be quite a radical message. Reading past the inconsistent spellings we see the names of Tamar, Rahab, Ruth and Bathsheba, four women mentioned in the first six verses. It is unusual for women to be mentioned at all in these genealogies, let alone women of such repute: a confrontational trickster, a Canaanite prostitute, a poverty-stricken Moabite, and an unfaithful wife.² Hardly the ancestry of kings.

And that's the point. Jesus did not descend from a pure, aristocratic bloodline. He came to the world through the people to whom he would later preach his message of hope and the kingdom of God. Jesus could so easily converse with prostitutes, gentiles, sinners, the poor and the outcasts because those were his people, his forebears. Matthew's list heralds the coming of a very different king, and this opening tells us we are in for quite a story.

¹ As really next-to-nothing is known about Jesus's father we are here in the realm more of fiction than fact. The genealogy is possible, but unlikely; furthermore, given that the scriptures tell us Joseph was not actually the biological father of Jesus (God was) then the significance of Joseph's genealogy pales even further. What is interesting though is the numerical significance mentioned in verse 17: So all the generations from Abraham to David are fourteen generations; and from David until the carrying away into Babylon are fourteen generations; and from the carrying away into Babylon unto Christ are fourteen generations. Putting aside the fact that Matthew has omitted a few generations to make the numbers work, this gives us a total of 42 generations from Abraham to Jesus, curiously the number used by Douglas Adams in his novel 'The Hitchhiker's Guide to the Galaxy' as being the answer to life, the universe and everything. Make of that what you will.
² You can read about each of these women in the following reflections:
Feminine, Harlot, Scavenger and Lust.