For a start, if the lesson is over, I would start with "That" not "This".
As you get in the car at the beginning of the lesson: This is the first time I've ever driven.
At the end of the lesson: That was the first time I had ever driven.
In the piece you posted at the beginning, it said "Jack is driving the car". If that is the case and he is speaking while he is driving, then "This is the first time I have driven" is fine. That would be correct right up until the end of the lesson. After that, it was the first time he had ever driven.