Shortly before Jesus ascends into heaven, the apostles inquire if He would now "restore the kingdom to Israel." Sadly, this implies that they are still hoping for some sort of physical empire like David had (e.g., John 6:15), and Jesus' recent declaration about having "all power" certainly didn't help (Matt. 28:18). Jesus does not correct their misunderstanding at this time, but they will soon more perfectly comprehend the spiritual nature of the kingdom--with the enlightenment of the Holy Spirit that was to come! Additionally, their question implies their belief that the kingdom was yet to be established, and in this they are correct!
There is much evidence that can be studied to show when the kingdom of God would be established:
Can there be any doubt that the kingdom was not yet a reality at that point in time?
Furthermore, Jesus promised to build His church and give Peter the keys of the kingdom of heaven in Matthew 16:18,19. Certainly the "church" is "the kingdom of heaven" because Jesus would not build one thing and then give Peter the keys to something else! To give a person the keys to something is to give them the authority to both open it and admit in whom they will. Jesus gave Peter this authority regarding the church, and the apostles were all also promised this same right (cf. Matt. 18:18). It is easily observed that the apostles' actions (specifically Peter's) on Pentecost in Acts 2 and with Cornelius in Acts 10 opened up the kingdom of God to both Jews and Gentiles on the same terms.
Jesus' reply in Acts 1:7 that it is not their business to be concerned about the Father's timing reminds me of Deuteronomy 29:29 - "The secret things belong to the LORD our God, but those things which are revealed belong to us and to our children forever." It wasn't their place to know when the kingdom would come, although Jesus had already indicated that it wouldn't be long.
Acts 1:8 has been rightly called a summary of the book of Acts. Therein Jesus tells them four things they did need to know:
The power they would receive would enable them to work miracles in order to confirm the origin of the message they were speaking (cf. Mark 16:20; Heb. 2:3,4). This power would come when they were baptized with the Spirit (cf. Acts 2:1-4). Jesus explained in Mark 9:1 that the kingdom of God would come "with power" during that generation. The logical connection between Mark 9:1 and Acts 1:8 is clear: The kingdom (or church) would be established when the Holy Spirit came upon the apostles of Christ in a powerful way. Such is precisely what happened in Acts 2, which was a mere ten days after Jesus' ascension.
The apostles would be "witnesses" for Jesus. By definition, a witness is a person who has seen something firsthand and, on that basis, seeks to confirm a truth by his testimony. The apostles were true witnesses of the resurrection of Christ; they saw it firsthand. Jesus had prepared them for their great work as witnesses for over three years. Obviously, no one alive today can be a "witness" for Christ as they were.