The evangelism of the apostles revolved about three main notions: The Gift, The Hope, and the Promise. To often Christians see these as merely vague terms [or they assume they all mean "life in heaven after I die."]
Putting a fine point on these terms allows not only to interpret individual passages better, but also grants a richer understanding of what apostolic evangelism was about.
When the apostles speak of "Hope," they do not refer exactly to "eternal life in heaven for believers." The hope they refer to is the idea that what God had done to Jesus already [bodily resurrection], the Almighty would do to everyone later.
To understand why this Hope is so exclusively Christian, you have to understand the culture of 1st century AD. Some Jews believed in a bodily resurrection that would occur far in the future, and many Jews did not. Those outside Judaism generally did not believe in a bodily resurrection at all.
What no one believed was that the Messiah would come, die, and be resurrected before everyone else. Not even Jesus' disciples understood that [which is why they deserted Him... the Messiah was supposed to lead the Jews to victory over the Romans [and everyone else who had oppressed them: Luke 1:71 ] How could He do that if He were dead??
This is why they wonder in Mark 9:10 what Jesus means, since He cannot possibly mean He is going to literally die. It also comes through loud and clear in Luke 24:20-21, they had hoped (but no longer)...and what did they hope for, that the Christ would redeem Israel. Like David, they were waiting for Jesus to take His position as true King. Having Jesus die crushed these beliefs, for they did not understand He had to die [John 20:9, Luke 24:25-27]
So, the resurrection not only proves that the Christian God lives, but gave hope in their own resurrection later. Note the wording of Acts 4:2 and Acts 17:32 --- this was the central message of their evangelism (as well as Jesus as the Christ and Jesus as Judge, see post on evangelism in Acts.) Paul also avers that belief in resurrection of the dead is absolutely required of believers [1st Corinthians 15:12-14].
This Hope in the resurrection of the dead was the main reason Paul was in so much hot water in Acts. It was, after all, directly opposed to the beliefs of the ruling sect of Judaism. [Acts 23:6, Acts 24:15, Acts 24:21, Acts 26:6-8.]
People use the term "a free gift" often in evangelism today, but for Paul and the other apostles, the term had a different meaning. The Gift is nothing other than the Holy Spirit.
Jesus uses this to refer to the Holy Spirit in John 4:10, Peter refers to the Spirit as "The Gift" three times: Acts 2:38, Acts 8:20, and Acts 11:17. Luke uses the term in this way in Acts 10:45, and Paul does so in 1st Timothy 4:14, and 2nd Timothy 1:6. The writer of Hebrews follows suit in Hebrews 6:4.
[Note it is important to separate "The Gift" (with the "the") from situations where there is no "the," also there are 3 words for "gift" common in the NT, and only 2 of them appear to be used in this way, the other is more of a term for "offering."]
The (Promised) Promise
And now we come to an interesting term. "The Promise." We know that God promised Abraham to bless the world through his seed, but how was God going to do that? Peter answers this question for us as well, in Acts 2:33. This is the promise Jesus refers to in Luke 24:49, and in Acts 3:26, we find that it is, in fact, the fulfillment of the promise to Abraham.
The Holy Spirit is further referred to as "The Promise" in Acts 2:39, Galatians 3:14, Ephesians 1:13, Ephesians 3:6, Hebrews 9:15-(The writer views us as receiving the Holy Spirit as an inheritance from Christ. Note this is definitely referring to the Holy Spirit available now that was not available earlier before Christ died: see Hebrews 9:16, Hebrews 9:8, and, most notably Hebrews 11:39-40, where the Spirit is once again referred to as what was "promised.")
But, the Holy Spirit is not only the fulfillment of a promise, it also acts as a promise...a reminder of the full salvation available when our bodies are transformed and New Jerusalem arrives [ Revelation 21:2]. The Holy Spirit, in granting us power over the desires of a flesh set against God [Romans 6:6] grants us a slice of our transformed future. It allows us to already begin living the life of the next era today. Indeed, it calls us to do so, for we are no longer to live for ourselves, but rather live for Christ.
In this way, the Holy Spirit is then a Promise itself. Not on an individual basis wherein we "know we are going to heaven because we have the Spirit." The Spirit allowed those in Matthew 7:22-23 throw out demons and prophesy in Christ's name, but it did not see them through the Judgment, and Hebrews 6:4-6, Hebrews 10:26-27, and 2nd Peter 2:20-21 all describe that the Spirit is not a personal guarantee but a global one. God will not be mocked.
Understanding "The Promise" is crucial to getting a handle on Paul's letters to the Galatians and Romans, where the term is used very often.