The Gospel according to Luke and the Acts of the Apostles are two different books of the Bible, but really, they are two volumes of the same book. The author of both is Luke, who was a physician and companion of the Apostle Paul. Luke's Gospel account is a carefully researched and arranged account of the earthly life and ministry of Jesus Christ—his birth, life, ministry, death, resurrection, and acsension— and Acts is an account of how Jesus is still at work through the Christian Church.
Luke isn't named in these volumes as the author (and so they are "formally anonymous") but we're confident that he is. There is evidence within the books (e.g. Acts 28:1-14 - notice the "we" language), from other books of the Bible (e.g. Philemon 24, Colossians 4:4), as well as writings from the early church fathers (e.g. Justin Martyr, Irenæus, Tertullian) that it was Luke.
Luke introduces his Gospel account with a dedication to a man named Theophilus (and references him again as he opens Acts (Acts 1:1-2), one of the reasons we consider Luke-Acts to be two volumes instead of two separate books). In Luke 1:1-4 we learn that Theophilus had been taught something about the Christian faith, and Luke had carefully followed and written down the account so that Theophilus could be certain of what he had been taught. So we can view Luke-Acts as an account—a narrative—intended not only to share the Gospel of Jesus with those who've never heard it, but also to help strengthen the faith of those who have.
Some themes of Luke-Acts:
The "Shape" of of the Journey Think of an hourglass. We start journeying at the top, and like the sand, we travel through the narrow part then out to the wide part again. Luke begins with a wide view, talking about all that has happened (Luke 1:1) and how he's put together an orderly account of it. The action starts fairly wide, with a picture of Israel as a nation being ruled by Herod (Luke 1:5). The account then "narrows" towards Jerusalem (Luke 9:51 is a very important verse), with the death and resurrection of Jesus as that narrow focal point. The account then widens out again as the Church spreads the Gospel after the day of Pentecost. Acts 1:4-8 summarizes this second "half" of the account.
God's Plan We often find Jesus speaking of necessity and purpose in Luke's Gospel account. Jesus knows that the things that happened needed to happen that way, so that the salvation of the world would be accomplished the way that God desired it to happen. (Examples: Luke 4:42-44, Luke 9:21-22, Luke 13:31-33, Luke 22:37, Luke 24:4-7, maybe most importantly Luke 24:26 and 44-47).
Jesus at Work We clearly see Jesus at work in Luke's Gospel account. But Jesus is still at work in Acts, following His ascension (which "glues" the accounts together: Luke 24:44-52 and Acts 1:1-8). Jesus works through the Church, which carries on the Gospel into all the world.