To which people did Jesus bring the “good news”?

I am confused about two places in Matthew where Jesus is instructing his disciples where to spread the gospel message – Matthew 10:6 and Matthew 15:24. Was Jesus calling them to bring the good news to only the "lost sheep of Israel"?

In Matthew 10, Jesus instructed His disciples to bring the news of the Kingdom to Israel only:

Matt. 10:5 These twelve Jesus sent out after instructing them: “Do not go in the way of the Gentiles, and do not enter any city of the Samaritans;
Matt. 10:6 but rather go to the lost sheep of the house of Israel.
Matt. 10:7 “And as you go, preach, saying, ‘The kingdom of heaven is at hand.’
Matt. 10:8 “Heal the sick, raise the dead, cleanse the lepers, cast out demons. Freely you received, freely give.
Matt. 10:9 “Do not acquire gold, or silver, or copper for your money belts,
Matt. 10:10 or a bag for your journey, or even two coats, or sandals, or a staff; for the worker is worthy of his support.

Jesus was the Jewish Messiah, and as God promised in His covenants with Israel, the Messiah was sent to to establish an eternal Jewish Kingdom on earth. In Chapter 10 of Matthew, that offer of the Kingdom went to Israel alone in keeping with the covenants, so Christ ordered that the Gospel must be shared only with the Jewish people at that time.

Later, in Chapter 12 of Matthew, the leaders of Israel committed the unpardonable sin of rejecting the Messiah and His Kingdom, and then from that moment the offer of the Kingdom was withdrawn from Israel. Thereafter, Jesus began to train His disciples to reach the Gentile world.

For more information on this critical moment in Jesus’ ministry, please read What is the unforgivable sin?

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So in Chapter 15, Jesus tests the faith of a Gentile woman:

Matt. 15:21 Jesus went away from there, and withdrew into the district of Tyre and Sidon.
Matt. 15:22 And a Canaanite woman from that region came out and began to cry out, saying, “Have mercy on me, Lord, Son of David; my daughter is cruelly demon-possessed.”
Matt. 15:23 But He did not answer her a word. And His disciples came and implored Him, saying, “Send her away, because she keeps shouting at us.”
Matt. 15:24 But He answered and said, “I was sent only to the lost sheep of the house of Israel.”
Matt. 15:25 But she came and began to bow down before Him, saying, “Lord, help me!”
Matt. 15:26 And He answered and said, “It is not good to take the children’s bread and throw it to the dogs.”
Matt. 15:27 But she said, “Yes, Lord; but even the dogs feed on the crumbs which fall from their masters’ table.”
Matt. 15:28 Then Jesus said to her, “O woman, your faith is great; it shall be done for you as you wish.” And her daughter was healed at once.

Jesus has traveled to a Gentile region and encountered a Canaanite woman. While Jesus had been unable to convince the Jewish leaders He was their Messiah, here He finds a Gentile woman who would not retreat from her confidence that Jesus was the Messiah. Jesus tested her faith by asking her wasn’t the Jewish Messiah sent only to the nation of Israel?

Only a person truly moved by the Spirit of God to understand the promises of God in the Old Testament concerning the Jewish Messiah would have been able to appreciate Jesus’ question and respond properly. Therefore, Jesus’s question was an excellent test of the woman’s motives in wanting healing. Her answer reflected an appreciation of God’s plans to graft Gentiles into the promises given to Israel.

Therefore, the woman’s response reflected her faith in the word of God and His promises concerning Israel. Because of her faith, Jesus healed her. She is the first Gentile in the Gospel to demonstrate faith and be healed, which was proof that the Gospel was now moving outward to reach Gentiles as well. This change was in response to Israel's unpardonable sin.