Did Jesus say to preach to Gentiles, or not?

If Jesus meant for his disciples to go to all nations and preach the gospel to Gentiles, why did Peter need a vision in Acts 10 to visit Cornelius, a Gentile (actually a proselyte of the gate)? Why did Peter tell Cornelius it was an unlawful thing for him, a Jew, to be in the house of a Gentile?

Because Jesus in His earthly ministry was strictly preaching to "the lost sheep of the house of Israel". Salvation to the world, apart from the nation of Israel, was not revealed until the ascended Christ shared this "mystery" with the Apostle Paul. The apostle Paul is THE apostle of the Gentiles (ROM 11:8 KJV). The "mystery" revealed is revealed in Ephesians 3. Rightly divided the word of truth (II TIM 2:15) and you will see Paul's unique authority and apostleship.

Did Jesus say to preach to Gentiles, or not?

Most of the preaching Jesus did was to Israel announcing that the kingdom of heaven was at hand.

Matthew 4:17 From that time Jesus began to preach, and to say, Repent: for the kingdom of heaven is at hand.

This was because Israel had been promised a king and a kingdom and Jesus came to offer that kingdom to Israel with a new covenant.

Jerimiah 31:31-32 Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD:

Nothing Jesus ever said hinted that he knew Israel would drop the ball and that the kingdom would be delayed for over 2,000 years. In the “Great Commission” Jesus tells the disciples what the expected role for Israel is to be.

Mark 16:15-18 And he said unto them, Go ye into all the world, and preach the gospel to every creature. He that believeth and is baptized shall be saved; but he that believeth not shall be damned.
And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues; They shall take up serpents; and if they drink any deadly thing, it shall not hurt them; they shall lay hands on the sick, and they shall recover.

As far as the gentiles go, we have a hint that they were to be included.

John 12:20-24 And there were certain Greeks among them that came up to worship at the feast: The same came therefore to Philip, which was of Bethsaida of Galilee, and desired him, saying, Sir, we would see Jesus. Philip cometh and telleth Andrew: and again Andrew and Philip tell Jesus. And Jesus answered them, saying, The hour is come, that the Son of man should be glorified. Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.

From this we might conclude that Jesus was saying that there was a harvest to be had among the gentiles, but that it had to wait until his resurrection.

Yes, He talked about it after His resurrection:

Go ye therefore, and make disciples of all the nations, baptizing them into the name of the Father and of the Son and of the Holy Spirit:

Teaching them to observe all things whatsoever I commanded you: and lo, I am with you always, even unto the end of the world. Matthew 28:19-20 ASV

Jesus told the disciples to preach the Gospel to all nations.

Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit (Matthew 28:19, ESV)

But you will receive power when the Holy Spirit has come upon you, and you will be my witnesses in Jerusalem and in all Judea and Samaria, and to the end of the earth.” (Acts 1:8, ESV)

Paul wasn't the first one nor the only one who preached the Gospel to the Gentiles. According to Acts, Peter was the first to preach the Gospel to the Gentiles. Peter received a vision from the Lord and received a command to preach to the Gentiles in the house of Cornelius the gentile. The Gentiles were filled with the Holy Spirit and were baptized.

While Peter was still saying these things, the Holy Spirit fell on all who heard the word. And the believers from among the circumcised who had come with Peter were amazed, because the gift of the Holy Spirit was poured out even on the Gentiles. For they were hearing them speaking in tongues and extolling God. Then Peter declared, “Can anyone withhold water for baptizing these people, who have received the Holy Spirit just as we have?” And he commanded them to be baptized in the name of Jesus Christ. Then they asked him to remain for some days. (Acts 10:44-47, ESV)

Peter reported this to the believers in Jerusalem and they all glorified God.

When they heard these things they fell silent. And they glorified God, saying, “Then to the Gentiles also God has granted repentance that leads to life.” (Acts 11:18, ESV)