The first place to look would have to be Paul. He was the most prolific soul-winner and writer of the early church. According to Liberty University Professor Gary Habermas: "On matters concerning the historical Jesus, Paul was an authoritative source, an eyewitness who was close to the data he records." If early Christians required evidence, there is no way that Paul could have been successful if he did not have the goods. He must have talked to all the eyewitnesses and checked out their stories, right?
Amazingly, Paul seems to say quite emphatically that he did not do any sort of investigation before he converted (certainly nothing as impressive as Lee Strobel's two year quest). Paul tells us in Galatians 1:11-18 that he got what he knew directly from Jesus Christ.
I want you to know, brothers, that the gospel I preached is not something that man made up. I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ. For you have heard of my previous way of life in Judaism, how intensely I persecuted the church of God and tried to destroy it. I was advancing in Judaism beyond many Jews of my own age and was extremely zealous for the traditions of my fathers. But when God, who set me apart from birth and called me by his grace, was pleased to reveal his Son in me so that I might preach him among the Gentiles, I did not consult any man, nor did I go up to Jerusalem to see those who were apostles before I was, but I went immediately into Arabia and later returned to Damascus. Then after three years, I went up to Jerusalem to get acquainted with Peter and stayed with him fifteen days. I saw none of the other apostles—only James, the Lord's brother.
Of course, Paul's vision on the road to Damascus was probably pretty darn impressive, but it is not the kind of evidence that could have been verified by anyone who heard Paul's preaching.
If the only evidence Paul had when he began his preaching was this visionary experience, then the people who heard Paul must have relied upon that vision when they converted. They did not have the testimony of the original apostles who were eyewitnesses to Jesus' life and ministry because Paul did not have the testimony of the original apostles available to him. At least for the first three years, Paul's testimony to his own vision was more than sufficient to convert large numbers of pagans.