As for Paul, the good news begins with God, not with Jesus.
The gospel (good news) is about or from God.
It is also God's good news that he intervenes in the world.
Jesus is the Son of God who demonstrated God's good news through faith.
Jesus's work of God constitutes "the gospel of Christ."
However, I often see theological schools' mission statement that elevates only "the gospel of Christ." Of course, what the gospel of Christ means is a bone of contention.
Now Christians are those who follow Jesus and his faith. They are set right with God. They must bring good news to all as "a letter of Christ" (2 Cor 3:3).