What are the "even greater things" Jesus spoke of in John 14:12?

Right from the start we read in Acts: "A deep sense of awe came over them all, and the apostles performed many miraculous signs and wonders" (2:43). The supernatural was always meant to be part of normal Christian outreach. And it was not just leaders who were expecting them.   

"All the believers lifted their voices in prayer to God … Give us, your servants [slaves], great boldness in speaking your word. Stretched out your hand with healing power; may miraculous signs and wonders be done through the name of your holy servant Jesus" (4:30).

And let's not miss the purpose of the signs. "The Lord worked with them and confirmed his word by the signs that accompanied it" (Mk 16:20). "The Lord confirmed the message of his grace by enabling Paul and Barnabas to do miraculous signs and wonders" (14:3).

Very soon the early church was replicating what Jesus had done: "Crowds came from the villages around Jerusalem, bringing their sick and those possessed by evil spirits and they were all healed" (5:15). The same is recorded of Jesus in Mat. 8:16.

Greater "things" is the literal meaning; not specifically "miracles". I think this is so we don't restrict what God wants us to aim for! One commentator writes: "The works of the apostles after the resurrection were not greater in kind than those of Jesus, but greater in their sphere of influence and extent". Another has: "Jesus never saw multitudes converted within minutes as Peter did". And still another says: "There are many miracles of healing in Acts, but the emphasis is on the mighty works of conversion."

All this is true, but, in addition, I felt God say to me (so this needs to be weighed as I couldn't find a commentary that made the point) that the "things" could be seen as "greater" in that they would be accomplished through flawed human beings – works in progress – with far from perfect faith and never to live completely without sin in this life.

Later in the story, even Peter himself had to be publicly rebuked by Paul for his hypocrisy in withdrawing his fellowship from the Gentile believers. This was an extremely serious matter as it confused the issue over whether or not Gentile converts should be circumcised and obliged to obey the Law of Moses. It even pointed to a gospel of works rather than a gospel of grace and faith. Peter was flawed like the rest of us, but he was also a man of outstanding humility, faith and power that brought much glory to his precious Saviour.