I would like to introduce my faith to a non believer through reading a Gospel. Which one should I start with and why?
You are right in saying that, in introducing the bible to a non-Christian, the Gospel is a good place to start, since it is the life and words of Jesus Himself.
As to which of the four gospels is a good choice, my opinion is that there is really not one that is better than another. You can tell your friend to just start with one of them.
But just for completion, the following are some of the most obvious characteristics of each.
This gospel seemingly was written by and for Jews. It is replete with asserting that Jesus is the fulfillment of the Old Testament law and prophecies. It is also set in the tone of a Jewish teacher, with extensive teachings that are unique to this gospel. Its text therefore also contains numerous Jewish expressions that may seem foreign to us.
This is the shortest of the four, and is action-packed. It is written by a Jew with limited Greek and so the original text has the most grammatical problems. Many scholars believe Mark is the earliest of the gospels. The whole gospel is written as in one go, and you can read the whole thing as a non-stop action easily in 3 hours. However, it does contain a lot of details not found in other gospels, as if Mark was the first to capture them. This is perfect for someone who just wants to get a quick grasp on the life of Jesus.
Luke is quite similar to Matthew, with many passages that are only common to them. But Luke is obviously written by a Greek for Greeks, and has the best Greek usage of the four. Luke is unique in emphasizing Jesus’ humanity and compassion, as well as a significant emphasis on the role of the Holy Spirit throughout the ministry of Jesus. Luke also contains the most moving parables of Jesus like the prodigal son and the good Samaritan.
John is the last gospel to be written, with the assumption that the readers had already read one or more of the other three. Unlike the other three, which are very similar in narrative and style, John is very different. Its language is so simple that a grade schooler can read, yet the deeper meaning can confound the most learned. It is a book of mysticism, yet it is painstakingly accurate in the fine details of history and culture. Scholars have found that if there are any discrepancies between John and the other three gospels, almost 10 out of 10 John is the accurate one. John’s gospel is not merely a narrative, but a deep reflection and revelation of the depth of who Jesus is.