Peter’s sermon is a summary of the sorts of things he would have preached in any similar context. He is speaking to rather well educated Jews in the Temple, people who knew their Hebrew Bible very well. Rather than pursue modern logical arguments, he turns to the Psalms and shows that David does not exhaust the meaning of the text. Since the messiah is to be a new David, the psalms Peter cites are turning into prophecies of Jesus’ resurrection.
In order to show that the Messiah would rise from the dead, Peter quotes Psalm 16:8-11. In this text, David states his faith that God will not abandon him in the grace not “allow him to see decay.” Peter points out that David died and was not resurrected, his tomb was still venerated in Jerusalem to that day. Perhaps people in the audience had already visited the tomb of David during their visit to the City! (Modern tours of Israel often visit the Upper Room and the Tomb of David at the same time since they are relatively close together.)
Psalm 16 is remarkable in that both Peter and Paul cite it as a prophecy of the resurrection of Jesus (cf., Acts 13:32-37). Yet when one reads Psalm 16, there is little there that hints at a messianic interpretation. To tease out a messianic implication from the psalm Peter blends it with Ps 132:11 and applies it to Jesus.
To me, this is an exegetical maneuver which I would not a student to make, and probably if I heard a pastor use Scripture in this way I would probably have a few things to say about his exegetical method. But int he context of Jewish interpretation of Scripture, this makes sense. Combining texts in this way creates new possibilities which are then applied to new situations. I think this might be a case where we should be careful how we try and apply Scripture, Peter is not giving a lesson on how to read the Hebrew Bible, only showing that these texts allude in some way to the resurrection.
To further his case, Peter cites Psalm 110, another well known messianic prophecy. There David is told that he would be exalted to the very throne of God and that God would make all his enemies his footstool. This too cannot have been exhaustively fulfilled in the life David. Although David was given great victories, and he was the greatest king in Israel’s history, he was not raised to the level of the throne of God!
Peter therefore tells the crowd that Jesus non only rose from the dead but was taken up to heaven like Elijah or Moses (or Enoch, for that matter). In those three cases, the person was a highly respected prophet who did not experience death. Like the great men of old, God confirmed Jesus’ message by doing miracles through him, but he allowed him to die in order to initiate the new covenant.
Since Jesus fulfills the psalm which David could not, he is confirmed as the Lord and Christ (Acts 2:36). This is the most shocking point in the whole sermon – everything which the Hebrew Bible looked forward to had happened with Jesus, he was in fact the Lord and Messiah.