The Book of Acts is an integral part of the New Testament because, in many ways, it is both a transitional book and an instruction manual.
First, the Book of Acts serves as a transitional book between Jesus' life, ministry, and Resurrection to His followers and the witnesses left behind. In other words, this is the first time we see what living the life Jesus preached looks like without His actual presence.
The first eleven verses of the book recount Jesus's ascension and then says this, in verses 12-15:
Then they returned to Jerusalem from the Mount of Olives, which is near Jerusalem—a sabbath day’s journey away. When they entered the city, they went to the upstairs room where they were staying. Peter, John, James, and Andrew; Philip and Thomas; Bartholomew and Matthew; James, Alphaeus’ son; Simon the zealot; and Judas, James’ son—all were united in their devotion to prayer, along with some women, including Mary the mother of Jesus, and his brothers. During this time, the family of believers was a company of about one hundred twenty persons.
This, then, is how it begins. Jesus ascends, and they begin their journey together, the one hundred and twenty people who will change the world by teaching others what Jesus taught them. The primary players are named and we will hear from or about them again, especially Peter and Paul, as their journey continues.
The Book of Acts is also a book of transition because it moves us from one form of writing, the Gospels (the story of Jesus), to another, the Epistles (letters). This book sets the stage, then, for the followers of Christ to spread the Gospel of Jesus both in person and through letters. It paves the way for the relationship-building that is going to grow the church.
As an instruction manual for the church, the Book of Acts sets forth both the basic tenets of the faith and many of the principles by which followers of Christ should live. The tenets of the faith, in part and as outlined by Paul in this book, include: salvation through the forgiveness of sins, being baptized as the sign of a new creation, and receiving the gift of the Holy Spirit (sustaining power). The principle way of living by the church, in part, includes these elements from Acts 2:42-47:
The believers devoted themselves to the apostles’ teaching, to the community, to their shared meals, and to their prayers. A sense of awe came over everyone. God performed many wonders and signs through the apostles. All the believers were united and shared everything. They would sell pieces of property and possessions and distribute the proceeds to everyone who needed them. Every day, they met together in the temple and ate in their homes. They shared food with gladness and simplicity. They praised God and demonstrated God’s goodness to everyone. The Lord added daily to the community those who were being saved.
Without the Book of Acts, there would be a tragic gap between the Good News of Christ and the existence of the church, which is evident by the fact that the Epistles are sent to already established churches. It is a crucial book of the Bible because it connects what was (Jesus's ministry and teaching) to what will be (pastors and laymen preaching and teaching the Gospel of Christ).