“Good News of God.” The words “of the kingdom” were added by copyists, to conform this verse to many others that appear in the Four Gospels. This is the only use of “Good News of God” in the Gospels, and from the context it is clear that it does indeed refer to the Good News of the Kingdom.
“Jesus came into Galilee.” In this context, the Word saying, “Jesus came into Galilee” refers to his “coming” as the fulfilment of a divine call to preach the Good News there. In this context, “came” (or the Greek could also be translated “went”) does not seem to refer as much to a change in physical location, (although it does do that) as to the fact that Jesus is following the leading of the spirit. He had been in Galilee before this. After Jesus had been baptized by John and spent 40 days in the wilderness, he had met disciples (John 1:35-51) and then returned to Galilee for a wedding at Cana (John 2:1-11) then gone to Capernaum (John 2:12) and even gone back to Jerusalem for Passover (John 3).
John the Baptist said one would come who would baptize in holy spirit (Mark 1:7). Jesus came from Nazareth to John (Mark 1:9). Jesus came into Galilee (Mark 1:14). It is likely that in this context Jesus was coming up from Jerusalem to Galilee.