A key word in the apostolic message of the Gospel is "believe." A common refrain is to "believe in the Lord Jesus Christ," or something similar. Brief thoughts on what this means:
Background: John the Baptist and Jesus both show up foretelling imminent judgment on Israel in connection with the coming of the kingdom of God. The "kingdom of God" picks up an Old Testament theme: the world is under the dominion of pagan gods through human rulers who worship them. God chooses a people, Israel, through whom He will reclaim the His own dominion. God will bring the nations, currently allegiant to idols, back under His kingship, ergo the kingdom of God. That this kingdom is coming means that God is initiating the climactic stage of this project.
Before God can judge the pagan world order, however, He must clean up His own house. His people are not presently fit to be the instrument of His renewed reign, so He will first have to prune them and refine them with fire. John and Jesus' initial audience were being asked to believe their message (1) that God's kingdom climax was underway, (2) that this would first involve a purifying judgment on God's people, and (3) that those who wished to share in this coming victory of God would need to embody the righteous requirements of the Torah in spirit with a whole heart.
Beyond this, the kingdom of God is associated with the heir to David's throne. The promised Messianic king will act as God's agent in reclaiming His reign over the pagan world. John does not seem to say much about this, but he does clearly expect some of his disciples to believe that the man he points out, Jesus, has at least some unique role to play in this process. Jesus' own preaching adds much more. He expects people to identify His own ministry with the coming of God's kingdom. On several occasions He expects people to take His own words, judgments, and deeds as equivalent to God's. And in the end He is clear that His audience must straightforwardly believe that He is the Messiah, the promised Savior.
With Jesus' death and resurrection, the apostles expanded the message's expectations. Now the hearers would need to acknowledge the coming reign, the coming judgment, the ethical standard, Jesus' status as Messiah, and the unexpected fulfillment of His messianic office by His death, resurrection, and exaltation. To believe the message is to acknowledge these acts of God for what they are, and the implication is that this naturally demands repentance and submission.
When the Gospel moves to the Gentiles, the character of the expected belief changes. The Gentiles are not expected to already be familiar with the rich covenant promises and narrative background of Jesus' work. The Gospel comes to them with the claim that all of their gods are idols, that the true Creator God is preparing to judge and repossess the nations, and that this will happen through the man He raised from the dead after crucifixion under Pontius Pilate, who has been installed as the Lord of the world. To believe in this Gospel is to acknowledge that the time of the pagan gods is up, that they will be overthrown, and that their human agents, like Caesar and other authorities, must either submit to Jesus as Lord in the name of the true God or be crushed.
The Gospel as given to the Jews called for those who claimed covenant with God to believe that He was about to judge His own people, establish His kingdom over the nations, and do these things through Jesus, His Son, the Messiah who died on behalf of Israel's sins and was raised and exalted to David's throne at God's right hand, and whose teaching would be the standard by which He people would be judged.
The Gospel as given to the Gentiles called for pagans to believe that their gods were idols, that the true Creator God was preparing to overthrow the idolatrous world order, and that God would do these things through Jesus, His Son, the king of Israel who was crucified under pagan rule but was raised and exalted to Lordship over the nations, whose ethical teachings would be the standard by which the nations would be judged.