Paul will have none of that. The gospel message is nothing less than power from God, for everyone who believes it, to be saved. Anyone, everyone, who puts their faith in Christ and his death for their sin on the cross will be justified—will be made right with God—and welcomed into God's family. If that is all true, and Paul had staked his life on it, then what is there to be ashamed of?
Paul continues that this salvation from God by faith in Christ is for the Jew first and also for the Greek. In this context, "Greek" is similar to the meaning of "Gentile," as a references to all the non-Jewish people in the world. Why for the Jew first? Paul will spend a lot of time on that concept in this letter. Jesus said simply that "salvation is from the Jews" (John 4:22). Paul's letter will show that God prioritized the Jews first in revealing that all can be saved by faith in Christ. Though they rejected that message as a nation, many Jewish people did indeed trust in Christ for their salvation, starting with Jesus' 12 Jewish disciples.
This explains in part why Paul often began his work in a new city by preaching in the Jewish synagogues before moving on to preach to the Gentiles (Acts 13:5). It's true that Christ sent Paul to preach to all the nations, but He also commissioned Paul to preach about Him to the children of Israel (Acts 9:15).