What are some effective ways to lead people into a closer relationship with the Lord? How were the disciples successful in their evangelism?
The most effective way is “faithfulness.”
David was faithful in shepherding a small flock of sheep. So God entrusted him with shepherding His people. He was faithful with the little that God had entrusted him with, (a few sheep in the wilderness). Therefore, he was entrusted with an entire army from Goliath. Following this, God entrusted David with His entire kingdom.
Saint Abba Rewis was not a deacon in the church, nor was he a monk, but he was a faithful layman. He is historically known for being faithful in his spiritual life and in his relationship with God. He became one of the saints of his generation, admired and loved by all those who knew him. I am sure this saint's faithfulness led many to the Lord Jesus Christ. Abba Rewis set an example of commitment for its own sake, commitment to the goal regardless of the means.
Saint Stephen was not one of the 12 Apostles. He served the church as a deacon. St. Stephen was so faithful even unto death that many, through him, came to accept Christianity.
Two other examples of faithfulness can be found in the Holy Gospel of St. Matthew 25: 21-23 (offering the two talents); and the Holy Book of I Kings 17:12-16 (offering the handful of flour).
Be faithful in what is within your capacity and then God will entrust you with what is beyond your capabilities.