The book of Acts narrates the foundation of the Church from its beginnings in Jerusalem through to an international movement. In Acts 1 Jesus explicitly links the expansion and growth of the Church with the coming of the Holy Spirit, and then the rest of the book then uses the coming of the Holy Spirit as evidence of what God is doing to grow the Church, to authenticate the Apostles' ministries, and to show the full acceptance of people into the church from all ethnic nations.
So in Acts 1:4-8, Jesus' very last words to his disciples before he ascended, he said:
Acts 1:4-8 (NIV): On one occasion, while he was eating with them, he gave them this command: “Do not leave Jerusalem, but wait for the gift my Father promised, which you have heard me speak about. For John baptized with water, but in a few days you will be baptized with the Holy Spirit.”
Then they gathered around him and asked him, “Lord, are you at this time going to restore the kingdom to Israel?”
He said to them: “It is not for you to know the times or dates the Father has set by his own authority. But you will receive power when the Holy Spirit comes on you; and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth.”
The disciples being Jesus' witnesses in Jerusalem, all Judea and Samaria, and to the ends of the earth forms a schema followed by the rest of the book of Acts.
So firstly we see the initial coming of the Holy Spirit at Pentecost in Acts 2. At this time many Jews from all surrounding areas had come back to Jerusalem for the festival. God sends the Spirit first to his disciples, who then go out into the city preaching and speaking in tongues. Many of the gathered Jews hear and come to faith, and three thousands people are baptised and join the Church that day.
Over the next few chapters the disciples continue to preach in Jerusalem. But after the execution of Stephen in Acts 7, the disciples are scattered throughout Judea and Samaria in Acts 8, however the Apostles remain in Jerusalem. The Jews considered the Samaritan people to be effectively half-caste, a mixture of Jew and Gentile. In the well known parable of the Good Samaritan we can see the contempt which most Jews would have felt towards the Samaritans. So what happens when the Gospel first goes to the Samaritans is very important for understanding the mission of God.
One of those who leaves Jerusalem is the deacon Philip. He goes to a city in Samaria, and preaches the Gospel, and many respond. Philip was a leader of the Church, but of secondary status compared to the Apostles. So when the Apostles hear that people in Samaria had accepted the Gospel, they sent two representatives, Peter and John to confirm the ministry of Philip. Many people consider that even amongst the 12 Apostles there was a smaller group of Jesus' closest disciples: Peter, James, and John. Peter and John are not just representatives then, but the absolute top of the church, humanly speaking. And so they go to Samaria, pray over the new believers, and the Spirit comes to them as well.
The Gospel going to the Samaritans is one thing; they still worshipped the same God as the Jews and tried to follow the Law of the Pentateuch (even if the Jews would say they did so incorrectly.) To fulfil Jesus' words in chapter 1 the Gospel would now go to the gentiles, to begin its journey to the ends of the earth. The Jewish traditions the Apostles were raised in demanded that the Jews separate themselves from gentiles. In Acts 10:28 Peter says "You are well aware that it is against our law for a Jew to associate with or visit a Gentile." There were a lot of prejudices that the early church needed to overcome, twisted misunderstandings of the Laws of purity which had been turned into racist exclusions of non-Jewish people. Even believing in the true God was not enough. That is what Cornelius was: in Acts 10:2 he and his family are called "God-fearers", a term indicating that they believed in the God of Israel, but had (probably) not been circumcised. To the Jews, and to a lot of the early church, the boundary for membership in the community was not belief and worship of God, but circumcision. Even Peter, leader of the Apostles, is still thinking in terms of Jewish requirements. It is only after God gives him a vision in which he is told to eat foods which the Jewish Law prohibited that he understood that everything had changed with the Gospel. So Peter goes to Cornelius, shares the Gospel with him and his household, and they believe. As they start speaking in tongues Peter's associates understand that the Holy Spirit has come on them too, and so he says that there can be no reason not to baptise them too. These uncircumcised Italians are now baptised members of the Christian Church.
Acts doesn't make explicit everything that we are meant to learn from it. We need to read careful to see what we should infer from it. To demonstrate what the spread of the Gospel should look like, God ordained that the Gospel would spread progressively, crossing ethnic boundaries in stages, and that Peter, first of the Apostles, would be there to witness and authenticate each event. We see the Gospel going to Jews, to Samaritans, and to gentiles, and each time the Holy Spirit is received by the new believers. And so that we can't misunderstand, the first gentile Christians are uncircumcised, and the Spirit comes to them before they are baptised by water. The Church would continue for many years to have debates over what was expected of gentile Christians, but these initial stories would be the foundation of gentile inclusion in the Church. After this time there is no hint that someone can repent and come to faith in the Gospel without the Holy Spirit coming to them. These were unique events to demonstrate and authenticate the expansion of the Church to all peoples.
There is one further weird story in Acts 19 where Paul comes across some "disciples" who had not received the Spirit. It's not completely clear, but it seems that they were disciples of John the Baptist, who may have heard something about Jesus but had clearly not heard the full Gospel. Paul tells them about Jesus and the Holy Spirit, re-baptises them in Jesus' name, and then the Spirit comes on them and they speak in tongues and prophesy. Rather than indicating that there can be Christians who haven't received the Spirit, I think this story just indicates that our language can be ambiguous, and that we should not just assume someone is a true disciple of Jesus. Just as Paul mistook John's disciples for Jesus' disciples, so too can we mistake cultural Christians for true followers of Jesus. Just like Paul in Ephesus, sometimes we need to evangelise the people in our churches so that they too can hear the Gospel and repent and believe.