From a much earlier blog in my history of writing blogs I touched on the topic of - What did Jesus say He came to do? (Blog Title: "What He Said About His Visit").  Here is an expanded version of that research.

Jesus said He came to bring division.
  • "Do you think I came to bring peace on earth? No, I tell you, but division." - Luke 12:51
Jesus said He came to do the will of The One who sent Him.
  • "'For I have come down from heaven not to do my will but to do the will of him who sent me.'" - John 6:38
Jesus said He came to claim lives that He would in turn not lose any that were given to Him.
  • "...that I shall lose none of all that he has given me..." - John 6:39
Jesus said He came to make a way to raise them up on the last day.
  • "...but raise them up at the last day." - John 6:39
Jesus said He came to provide eternal life.  You just have to trust in Him and believe in His offering of salvation.
  • "For my Father's will is that everyone who looks to the Son and believes in him shall have eternal life..." - John 6:40
Jesus said He came to fulfill prophecy.
  • "'Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them.'" - Matthew 5:17
Jesus said He came to make us "fishers of men".  Men and women who are involved in the lives of other men and women so as to attractively draw them into the Kingdom of God through belief in Christ Jesus' sacrifice.
  • "'Come, follow me," Jesus said, "and I will make you fishers of men.'" - Matthew 4:19
Jesus said He came to seek and draw the sinners unto Himself.
  • "'For I have not come to call the righteous, but sinners.'" - Matthew 9:13
Jesus said He came to be a disturbance.
  • "'Do not suppose that I have come to bring peace to the earth. I did not come to bring peace, but a sword.'" - Matthew 10:34
Jesus continued on this line of disturbance and division by saying His coming would create friction in the family.
  • "'For I have come to turn " 'a man against his father, a daughter against her mother, a daughter-in-law against her mother-in-law--'" - Matthew 10:35
Jesus said He came to be a servant.
  • "...just as the Son of Man did not come to be served, but to serve, and to give his life as a ransom for many." - Matthew 20:28
Jesus said He came "to preach the good news".  
  • "Jesus replied, 'Let us go somewhere else--to the nearby villages--so I can preach there also. That is why I have come.'" - Mark 1:38
Jesus said He came "to redeem His people".
  • "Praise be to the Lord, the God of Israel, because he has come and has redeemed his people." - Luke 1:68
The people declared that Jesus came as a great Godly prophet to "help His people".
  • "They were all filled with awe and praised God. "A great prophet has appeared among us," they said. "God has come to help his people." - Luke 7:16
Jesus said He came to bring salvation.
  • "Jesus said to him, 'Today salvation has come to this house, because this man, too, is a son of Abraham." - Luke 19:9
Jesus was confirmed as an awesome teacher.
  • "He came to Jesus at night and said, 'Rabbi, we know you are a teacher who has come from God. For no one could perform the miraculous signs you are doing if God were not with him.'" - John 3:2
Jesus was confirmed by the people and fulfilled the prophecy concerning The Great Prophet to come into the world.
  • "After the people saw the miraculous sign that Jesus did, they began to say, 'Surely this is the Prophet who is to come into the world.'" - John 6:14
Jesus said He came to provide abundant life or a fulfilled lifestyle.
  • "'The thief comes only to steal and kill and destroy; I have come that they may have life, and have it to the full.'" - John 10:10
Jesus said He also came to provide, as Himself, the world a light.  
  • "I have come into the world as a light, so that no one who believes in me should stay in darkness." - John 12:46
Jesus said He came to spoil the plans and accomplishments of the devil.  And He did.
  • "The reason the Son of God appeared was to destroy the devil's work." - 1 John 3:8