Luke 1:1–4 (NIV)
1 Many have undertaken to draw up an account of the things that have been fulfilled among us, 2 just as they were handed down to us by those who from the first were eyewitnesses and servants of the word. 3 With this in mind, since I myself have carefully investigated everything from the beginning, I too decided to write an orderly account for you, most excellent Theophilus, 4 so that you may know the certainty of the things you have been taught.
The main thing I'd like to point out in looking at Luke's opening paragraph is that while I believe all of the Bible is the inerrant, inspired Word of God, those who do not follow God do not think so. Even some of those who do follow the God of the Bible don't think this is so.
When talking to someone about reading Luke, I wouldn't try to prove that view point to them. Just present Luke as historical narrative. That's what Luke's opening paragraph is claiming to be. The NIV breaks this down into two sentences, but in the Greek, these four verses are one long sentence such as the ESV and NASB present it.
Luke was one of many, according to him, who wrote about Jesus and the things that happened. Luke points out in verse 2 that the information was handed down by eyewitnesses and servants of Jesus. Eyewitnesses. As I said, Luke is writing a historical narrative based on eyewitness testimony. Leave it at that when talking with friends. If Luke says Jesus said "X", then Jesus said "X". If Luke says Jesus did "Y", then Jesus did "Y".
Luke tells us in verse 3 that he carefully investigated. Many people have tried to discredit the Bible have found Luke to be reliable. Josh McDowell, Gary Habermas, William Ramsey, Frank Morrison. All men who set out to prove the Bible false and after careful investigation ended up following.
It is thought that Luke was perhaps a slave and that Theophilus was his owner. Luke's gospel is a part one of a two part series. If you read the opening verses to the book of Acts you'll see that it is Luke's second volume.
Acts 1:1–3 (NIV)
Jesus Taken Up Into Heaven
1 In my former book, Theophilus, I wrote about all that Jesus began to do and to teach 2 until the day he was taken up to heaven, after giving instructions through the Holy Spirit to the apostles he had chosen. 3 After his suffering, he presented himself to them and gave many convincing proofs that he was alive. He appeared to them over a period of forty days and spoke about the kingdom of God.