Acts was almost certainly written by Luke. Although he does not explicitly identify himself as the author, the opening verse strongly indicates that he was the author.
It is noteworthy that he says that in Luke he wrote about what Jesus began to do and teach. Jesus continues to do things, mostly through the church, and this continues to this day. We must never view ourselves as a purely human organisation, we are part of the body of Christ and He is the head.
Jesus gave several convincing proofs that He was alive. This gives lie to the idle notion some have that the early church were gullible people who would believe anything. They knew that in the normal course of events people do not rise from the dead, and so they needed proof that Jesus really had done so. Equally, God does not actually expect us to believe things without a good reason.
In the time between the resurrection and the ascension Jesus was teaching and preparing His disciples about the kingdom of God. He had spoken to them about these things before, but they didn't really understand. Now that He had risen from the dead they could begin to truly understand. Likewise, it can take events in our life to enable us to fully understand certain aspects of God's word.
He told the disciples to wait for the Holy Spirit. The church is nothing without the Holy Spirit. He is the One who directs and empowers us.