"Brothers, children of Abraham, and you God-fearing Gentiles, it is to us that this message of salvation has been sent." NIV translation
It is almost as if you can feel Paul pleading with the people to understand about salvation through Jesus Christ. This verse can serve to remind us that this good news is for all men who will accept it.
"The people of Jerusalem and their rulers did not recognize Jesus, yet in condemning him they fulfilled the words of the prophets that are read every Sabbath." NIV translation
Paul is telling the people that the words that they hear spoken in the synagogue each week refer to Jesus.
The lesson is for us as well. Many try to separate the Old Testament and the New Testament as if God just abruptly had to change his plans.
In reality, God knew it all from the beginning and Jesus, the Savior of the world, was not a "backup plan".
"Though they found no proper ground for a death sentence, they asked Pilate to have him executed. When they had carried out all that was written about him, they took him down from the tree and laid him in a tomb. but God raised him from the dead, and for many days he was seen by those who had traveled with him from Galilee to Jerusalem. They are now his witnesses to our people." NIV translation
Paul continues to show the people that Jesus is the promised Messiah. He reminds them that there was no legal reason to kill Jesus and that the grave could not hold him.
"We tell you the good news: What God promised our fathers he has fulfilled for us, their children, by raising up Jesus. As it is written in the second Psalm: 'You are my Son; today I have become your Father.'" NIV translation
Paul tells of how the words of the Psalmist are fulfilled in Christ and his resurrection. There is also a promise for us as, when we accept Jesus, we too become sons of God. We inherit all things through our relationship with the Father.