This is a very interesting question which many critics have debated endlessly. What becomes clear, however, if you read their ideas, is that the supposed recipient, whom Luke himself names in Luke: 1:4, Theophilus, is probably a name given for a general believer (it means friend of God) rather than a specific individual that both Acts and Luke were addressed to.
Critics seem to agree that the themes and what is included and excluded within Luke and Acts suggests that these two books were sent to a Gentile audience, given the importance of Gentiles in these books. The fact that Luke removes Jewish customs and language, in comparison with Matthew, for example, suggests that they were an audience with little knowledge about Jewish culture. The audience is familiar with the Old Testament in Greek, as Luke's frequent quotations are from the Greek version of the Old Testament. The name, Theophilus, suggests that the audience is already Christian.
Adding this altogether suggests that Luke's intended audience would be god fearing Gentiles by birth but with interest in the Jewish religion. Consider how Acts 10 for example includes prominent believers like Cornelius who would encourage such people to become followers of Jesus.