It's true (Sorry to repeat myself but this is so important as it is the future for us all) Christ's government will start in the 1000 year reign and will continue after God makes the new heaven and earth. However, the first resurrection (or Catching away) is for only SOME of the dead and living, not everyone. We are told some information about the first Resurrection group:
Revelation 20:4
4 I saw thrones on which were seated those who had been given authority to judge. And I saw the souls of those who had been beheaded because of their testimony about Jesus and because of the word of God. They had not worshipped the beast or its image and had not received its mark on their foreheads or their hands. They came to life and reigned with Christ a thousand years. 5 (The rest of the dead did not come to life until the thousand years were ended.) This is the first resurrection. 6 Blessed and holy are those who share in the first resurrection. The second death has no power over them, but they will be priests of God and of Christ and will reign with him for a thousand years.
The first resurrection group will be priests of God and of Christ and will reign with him for a thousand years on Earth. In Revelation 4:4 John saw thrones for those who have authority to judge, this refers to the 24 elders:
Revelation 4:4
Surrounding the throne were twenty-four other thrones, and seated on them were twenty-four elders. They were dressed in white and had crowns of gold on their heads.
12 of these Elders will be the Apostles who will judge the 12 tribes of Israel:
Matthew 19:28
Jesus said to them, "Truly I tell you, at the renewal of all things, when the Son of Man sits on his glorious throne, you who have followed me will also sit on twelve thrones, judging the twelve tribes of Israel.
John saw the souls of those who had been beheaded because of their testimony about Jesus and because of the word of God who had not worshipped the beast or its image and had not received its mark on their foreheads or their hands who come to life and reign with Christ for the thousand years. This refers to the 144,000:
Revelation 7:3-4
3“Do not harm the land or the sea or the trees until we put a seal on the foreheads of the servants of our God.” 4 Then I heard the number of those who were sealed: 144,000 from all the tribes of Israel.
144,000 divided by 12 is 6000 which is how many there is in a legion. Each of the 12 tribes will be judged by one Apostle and one Prophet of God making up the 24 Elders, as this verse tells us:
Ephesians 2:19-22
19 Consequently, you are no longer foreigners and strangers, but fellow citizens with God’s people and also members of his household, 20 built on the foundation of the APOSTLES and PROPHETS, with Christ Jesus himself as the chief cornerstone. 21 In him the whole building is joined together and rises to become a holy temple in the Lord. 22 And in him you too are being built together to become a dwelling in which God lives by his Spirit.
Along with the two Witness; Elijah and Moses, who will sit at Jesus' right and left, and along with the 24 Elders sitting around His throne, and the 144,000, this will make up Christ's government:
Isaiah 9:6
For to us a child is born, to us a son is given, and the government will be on his shoulders.
This government will rule over the Earth during the Millennial reign as it says in Revelation 20:5-6, and from the verses below we know that the first Resurrection will happen at Jesus Second coming after the tribulation:
1 Thessalonians 4;16-17
16 For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. 17 After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord forever.
At the end of the 1000 years after Satan is released and destroyed by God, the great multitude is resurrected:
Revelation 7:9
After this I looked, and there before me was a great multitude that no one could count, from every nation, tribe, people and language, standing before the throne and before the Lamb. They were wearing white robes and were holding palm branches in their hands.
Some will rise to everlasting life and go to Heaven, the rest to everlasting condemnation and go to Hell:
Daniel 12:2
And many of them that sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt.
This is when we will all receive our final eternal judgement and rewards:
Revelation 20:11-12
11 Then I saw a great white throne and him who was seated on it. The earth and the heavens fled from his presence, and there was no place for them. 12 And I saw the dead, great and small, standing before the throne, and books were opened. Another book was opened, which is the book of life. The dead were judged according to what they had done as recorded in the books.
The first resurrection is only for the 144,000 everyone else will have to wait 1000 years. After this Christ's government will continue to reign from heaven and over the new earth.