Below are some of the notes from Alex Kirk's sermon on chapter 1 of Jonah. There is a video at the end, as well as a link to the remaining notes.
1.This guy Jonah who we meet in the first verse is Jonah the son of Amittai and he actually appears in another part of the Scriptures In 2 Kings 14:25, Jonah of Amitai is referred to as a prophet and something that he prophesied came true, and so he gets referenced there as a real guy.
So if you're going to make up a story, you usually wouldn't make it up around a guy who really existed.
2. This reads like history. They had legends and stories in ancient Israel and this doesn't fit the pattern of how those works of literature worked. This read like history The other histories of other prophets and people in 1 and 2 Kings happening right around this same time sound much like Jonah's story
3. Jesus takes the story of Jonah seriously—he refers to this story saying that it was a foreshadowing of what he was going to do
Matthew 12:40 40 For as Jonah was three days and three nights in the belly of a huge fish, so the Son of Man will be three days and three nights in the heart of the earth.
In the end, obviously we each have to choose if we're going to believe that God could do something like this but I want to suggest to you at the very least this story is recorded and written down for good reason
And if we'll let it, we just might meet the God of the universe here.
Notes:
http://s3.amazonaws.com/churchplantmedia-cms/chatham_community_church_pittsboro_nc/week1-jonah1--sermon-manuscript-.pdf
Audio of sermon: