The history of Christianity for the apostolic era is based on the Gospels, the book of Acts in the Bible, references to the apostles in the early Christian writings, and Eusebius' Ecclesiastical History, written in A.D. 323.
You can read Jesus' story in the Gospels. There are no other contemporary sources for his life, other than a couple small references to the fact he existed in Josephus' Antiquities and Philo of Alexandria.
Note: I have an educated guess about Jesus' childhood and a link to some excellent research done by a man named Ray Vander Laan, who is to be commended, in my opinion, for an excellent use of his time on earth. We are all indebted to him.
Jesus was born, most likely, between 6 and 4 BC. This is based on Herod's reign, which ended in 4 BC.
I know, I know. That seems strange, since BC means "Before Christ." Our calendar wasn't invented until around A.D. 600. The fact that they were only off four to six years is impressive, actually.
Jesus died (and rose), most likely, in A.D. 29 or 30.
You can read the rest of Jesus' story in the Gospels.
John was the last of the apostles to die, ending the apostolic era. Irenaeus says repeatedly that he lived until the times of Trajan, whose reign began in A.D. 98. Since Irenaeus, who wrote around A.D. 185, was old enough to know people who knew John, his testimony carries a lot of weight.
Jerusalem was obviously the first church of the apostolic era. There's no evidence of any missionary activity, despite the "Great Commission" given to the apostles (Matt. 28:19-20), until Saul's persecution drove most of the Christians out of Jerusalem.
This led to the conversion of some Samaritans (Acts 8:5ff), an Ethiopian eunuch (Acts 8:26ff), and numerous Jews in Judea (Acts 8:4). God also used Peter to preach to the first Gentiles while he was visiting one of the Judean churches.
As time passed, the expansion that occurred under Saul's persecution led to churches being formed among the Gentiles (Acts 11:19-21). The most notable of these was in Antioch.
The apostles sent Barnabas to Antioch. By this time years had passed, and Saul was converted and living in his home town of Tarsus, not far from Antioch. Barnabas went and got him, and together they helped lead and build the church in Antioch, which is today the site of the patriarch of the Greek Orthodox Church.
After the Holy Spirit called Barnabas and Saul, soon to be Paul, as apostles ("sent ones") to preach throughout Greece, they formed several churches. Only four were formed on Paul's first missionary journey.
Paul was not an eye-witness of Jesus on earth. He met Jesus through visions. His letters give no indication that he had any familiarity with the Gospels nor any details of Jesus' life.
Paul would have spent between ten and fifteen years as a missionary before he died. They were busy years, but there were really only a few churches he formed. He stayed in Corinth for 18 months and Ephesus for 3 years. It was common for him to take time and build the churches he began.
Later, he would leave men like Timothy and Titus to build the churches until elders could be appointed (1 Tim. 3:1ff; Tit. 1:5ff; see sidebar).
Paul, as we know from Acts, wound up imprisoned in Rome. 2nd century Christians seem convinced that he was freed from that imprisonment and went west, possible even to Great Britain.
Peter, according to tradition, went to Rome (and possibly according to the reference to Babylon in 1 Pet. 5:13), and he brought Mark with him there. Eventually, Mark's Gospel would be accepted into the canon of the Church because Mark was Peter's companion.
Luke's was accepted by the church for similar reasons. He was Paul's companion.
Tradition has it that the other apostles spread widely during the apostolic era. Thaddeus went to Syria and Thomas to India. Several early Christians mention that Paul was released and went to Spain and England before returning to be martyred by the emperor Nero around A.D. 64.
Mark, though not an apostle, wound up in Alexandria.
The apostles established churches, taught those churches the Gospel and how to follow Christ, and then turned them over to appointed elders, whose job was to care for the Church and to safeguard the apostolic message, the Gospel of Christ, as the churches left the apostolic era.
That last point is extremely important. The apostles job was to give the message of Christ to the churches, and the churches to preserve that Gospel so that it could shine throughout the world.
It is this tradition that carries us out of the apostolic era and into our next era, the Pre-Nicene.