"When the islanders saw the snake hanging from his hand, they said to each other, 'This man must be a murderer; for though he escaped from the sea, Justice has not allowed him to live.'" NIV translation
When the native people saw Paul get bit by this poisonous snake, they immediately thought that it was a god at work bringing justice to him for wrongdoing. That is what happens even today as many people (even Christians) seem to think that God is just waiting to send punishment on the people every time they do something wrong. That is not the case for Christians as God does not condemn His children. Once we are saved by the grace of God, it brings freedom. This freedom also allows us to make mistakes without the fear that God is waiting to send a plague upon us.
"But Paul shook the snake off into the fire and suffered no ill effects. The people expected him to swell up or suddenly fall dead, but after waiting a long time and seeing nothing unusual happen to him, they changed their minds and said he was a god." NIV translation
When Paul was not killed, they changed their minds and tried to say he was a god. We see how quickly people change their opinions and how, if we chase after their approval, we will always be disappointed. What we actually see in this passage is the fact that God's people are not subject to death until He is finished with them here on the earth. Another example of this is found in the two witnesses described in Revelation 11. Paul was on a mission from God and no snake had the power to interfere with God's plan for him.
"There was an estate nearby that belonged to Publius, the chief official of the island. He welcomed us to his home and for three days entertained us hospitably." NIV translation
It would not have been normal for any official of the Roman Empire to entertain a prisoner but Publius must have heard about Paul and the snake. The number three is associated with the earthly expression of God's will and we see, here, that God's will was for Paul to visit with this official and be a witness to him.
"His father was sick in bed, suffering from fever and dysentery. Paul went in to see him and, after prayer, placed his hands on him and healed him." NIV translation
The man's father was sick and we see that God had sent Paul to heal him. It is interesting to note that before healing the man Paul prayed to God for guidance. This healing was important in showing Publius (a pagan) the power of the living God and so Paul looked for guidance from God. Today, it seems that it is a popular thing among "churches" to hold healing services and the like. Many times the people are lined up and herded through like cattle and often the man performing the healing is glorified instead of God. We can take a lesson from Paul as the people were lifting him up as a god (in verse 6) but he made sure that it was obvious that it was a work of the one true God and not of man.