The Story in Progress...
Acts is the second volume of Luke. The Gospel of Luke was about the things Jesus began to do and teach while he was on earth (Acts 1:1). The book of Acts picks up where Luke left off.
That is not to say that the ending of Luke and the beginning of Acts are exactly the same--more on that later. However, it is to say that Acts presumes the story of Jesus as it is told in Luke. The special emphases and themes of Luke are more or less the same emphases in Acts. At the same time, Acts does not necessarily assume the special features of Jesus' story in Matthew, Mark, or John.
What was Luke's presentation of Jesus like?  Luke of course shares the basic content of Mark in common with Matthew. Jesus preaches the kingdom of God (Luke 4:43). He heals people and casts out demons. He calls twelve disciples and trains them to fish for people. He gets into conflict with religious leaders and eventually is crucified before rising from the dead.
Luke also shares a significant amount of Jesus' teaching that it shares with Matthew alone. Luke's Sermon on the Plain (Luke 6) has a lot of the same material that is in Matthew's Sermon on the Mount (Matt. 5-7), including some Beatitudes, teaching on loving one's enemies, not judging, and building wising on rock. Luke has a lot of the teaching unique to Matthew's other sermons too.
But Luke has its own unique material as well. Luke alone tells us about the parents of John the Baptist and about Gabriel's appearance to Mary before Jesus was born. Indeed, one of the special features of Luke and Acts is the greater attention paid to the role women played in the Jesus story. In Luke we see it in stories about Mary and Martha and the Parable of the Persistent Widow. In Acts we will see it in Lydia and Priscilla, not to mention the way Peter's Pentecost sermon highlights how women as well as men will prophesy in the age of the Spirit (e.g., Acts 2:17).
Luke chose to begin the story of Jesus' ministry with Jesus reading Isaiah 61 in his home synagogue. This is the passage where Jesus connects his earthly mission to his message of good news for the poor. Luke more than any other gospel highlights Jesus ministry to the poor and gives his harshest words to the wealthy. Acts will continue this special emphasis with its picture of the early church sharing its possessions in common.
Some of the themes that we only catch a glimpse of in Luke become full blown in Acts. So Luke gives us clear hints that the gospel is for the whole world (e.g., Luke 2:32), but Acts will show this them in full bloom. In Luke we get a hint that Christians are not truly troublemakers, even if trouble follows them (Luke 23:14). Acts will make this point over and over again in its presentation of the early church.
Both Luke and Acts feature the centrality of prayer both to Jesus and his followers. Finally, while Luke predicts that the Spirit will come in force after the resurrection, Acts shows the fulfillment of the promise. If Luke gives us all the things Jesus began to do on earth, Acts gives us the things he continues to do through the Holy Spirit...
 For more details, see my book, Jesus: Portraits from the Gospels (Indianapolis: Wesleyan Publishing, 2013) **.