Chapter 1 of Luke's Gospel account is a rather long one: 80 verses (though, remember, chapters and verses aren't part of the original text; they came around a long time after the original writings). It begins with a formal introduction (Luke 1:1-4), establishing us in the fact that this is carefully researched and written so that a student of the Christian faith, a man named Theophilus (which means "friend of God", from Greek theos (God) and philos (brotherly love or friendly affection). Luke says that this is based on testimony of those who were called to spread the Word of God and who were eyewitnesses to the events (Luke 1:2).
After the introduction, Luke starts the narrative not with the birth of Jesus, but with John, Jesus' relative and forerunner of the Messiah. Luke shares about the announcement of the birth of John, who would become known as John the Baptist/Baptizer. Luke starts with the nation of Israel. The Hebrew priest Zechariah, through his work in the temple at Jerusalem (the city that will become the focal point of Luke-Acts later on), receives a word from God through an angel that he and his wife will have a son, though they were both older people and wouldn't expect to be able to have children (Luke 1:18).
Then we hear about Mary receiving the word of God also through an angel, that she would be the mother of God's Son (an event called The Annunciation, celebrated in the church year on March 25). Mary visits Elizabeth and they rejoice together. Mary's song, called "The Magnificat", is a wonderful song of praise to God for His blessings.
We then read about John's birth, which happened just as God said it would, and how Zechariah was released from his silence. He then blessed and praised God, and blessed his infant son John, who would grow up to be the one who pointed people to the Lamb of God who takes away the sin of the world.
In this chapter Luke is preparing us for the arrival of the Saviour, setting the historical and theological context for us so that he can proclaim the Saviour's birth.