Acts - is a transitional Book

The first eight chapters of Acts are predominately Peter preaching to the Nation of Israel about their condition. Remember they had sent their Messiah to the Cross, and Peter is not going to let them forget this. Then in Chapter 9 we have the conversion of the Apostle Paul, then in Chapter 10 we go back to Peter and the Jews in Jerusalem. And then after Chapter 15 Peter is no longer on the scene, and it's all Paul.

Peter and Paul did the same kind of signs, wonders, and miracles to confirm their apostleship (2 Cor. 12:12). These acts were seen as proof, by the Holy Spirit, that Jesus was the Messiah and the need for national repentance and obedience for the restoration and blessing of the nation. However, the Nation of Israel still rejected it. (Acts 28:25-28; Romans 11:25, etc.). So, God called out the Apostle Paul. 

I personal believe the transition finally changed over in 70AD when the temple was destroyed.  And Paul's gospel (1 Corinthians 15:1-4; Romans 10:9-10,13) went into full effect. So today, EVERYONE (Galatians 3:28) has to believe it to be saved.

Christ was crucified in 29 A.D. and of course 50 days after His Resurrection you have Pentecost. Then seven years later you have the martyrdom of Stephen in Acts Chapter 7. So, it was seven years from Pentecost till the death of Stephen. Then it's only one year later when Saul of Tarsus is converted on the road to Damascus. Then a year after Saul's conversion we find Peter in 38 A.D. going to the house of Cornelius in Acts Chapter 10. The Holy Spirit does this so Peter can come to Paul's defense in Acts Chapter 15.

Then in 40 A.D., which is three years after Paul's conversion, we find Paul has been to Arabia three years for seminary training with The Lord as his instructor. After that, for three years, from 37 A.D. to 40 A.D., he meets with Peter at Jerusalem, just for two weeks. Then Paul goes on up into his old Arabia of Tarsus. He ends up at Antioch and starts converting Gentiles to the Gospel of Grace. Then after he has been ministering up at Antioch and has been taking his Gospel to the Gentiles, the Jerusalem believers get all shook up and they call him and Barnabas down to Jerusalem for the counsel of Acts Chapter 15. And that Jerusalem counsel is also referred to in Galatians Chapter 2. This takes place in 51 A.D., which is about 22 years after the Cross. As you study all this verse by verse, you will see that at this point in time in Acts 15, Peter, remembering now what took place at the house of Cornelius, a Gentile, comes to Paul's defense.  

Acts 15:7 (KJV), "And when there had been much disputing, Peter rose up, and said unto them, `Men and brethren, ye know how that a good while ago (it had been 13 years) God made choice among us, that the Gentiles by my mouth should hear the word of the gospel and believe.'"

Now in that 13-year interval has Peter been going to Gentiles? No. He went right back to Jerusalem. He is still dealing with the Jew only, as you can see in Acts 11-19. But after Chapter 15, we find the Nation of Israel is falling away and the Gentiles are coming to the front, as Paul is starting the Gentile Church and the Age of Grace. Now Paul, of course, writes his letters within this time frame from approximately 46-49 A.D. to 67-68 A.D. when he is martyred. Then in 70 A.D. the Temple is destroyed. Now that's the Book of Acts, and it is 40 years in length. But always remember the first 7-8 years it's still God dealing with the Nation of Israel. But as you study just notice, they will slip completely off the scene, and the Apostle Paul will take the spotlight.