Hi Guojing,
I definitely agree that Christ innaugurated the Kingdom, He was the awaited for Messiah. Paul refers to this kingdom also! And yes, they mostly rejected him. Not all.
And yes we see the great commission progression through the book of Acts (in fact Acts gives us the road map Acts 1:8)
How did Jesus tell his people how they could be saved by Him in the gospels?
Paul never promised the Body of Christ, which is us, the same kingdom of heaven on Earth that was promised to Israel.
When the term
“the kingdom of God” appears in Paul's writings, it is a general reference to God’s domain, the realm of His influence (the righteous world system, Christians, and heaven).
This term can apply
both to redeemed Israel
and the Church the Body of Christ. The phrase
“the kingdom of God” is in contradistinction to the realm in which Satan and his cohorts operate (the evil world system, lost [unregenerate] mankind, hell, the lake of fire, fallen angels, et cetera).
A person enters
“the kingdom of God” when God rescues/saves him or her from Satan’s realm—having sins forgiven, being declared righteous before God, fellowship with God, able to participate in His will, not going to hell anymore, and so on. You can refer to Matthew 21:31, Luke 17:21, and Colossians 1:13.
Source:
https://forwhatsaiththescriptures.org/2015/06/24/kingdom-of-god-kingdom-of-heaven/
As for your 2nd question, Peter, thru the Holy Spirit, summed up how Israel could be saved by the gospel of the kingdom in Acts 3:19-21
19 Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord.
20 And he shall send Jesus Christ, which before was preached unto you:
21 Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began.
If you are reading it literally, without inserting what Paul taught us inside,
Peter is urging Israel to repent so that their sins may be, not surely, be blotted out.
And when will their sins be blotted out? In the future, when Jesus Christ returns for them, in his 2nd coming, the so called "times of refreshing".
This view by Peter is reaffirmed by what he wrote to the Jews in 1 Peter 4:17-18
17 For the time is come that judgment must begin at the house of God: and if it first begin at us, what shall the end be of them that obey not the gospel of God?
18 And if the righteous scarcely be saved, where shall the ungodly and the sinner appear?
Paul does not teach the Body of Christ that. We are forgiven the moment we believe in his death burial and resurrection for our sins.
It is not a may be, nor will it only come in the future.