The apostle John's visions, recorded in Revelation 20:7 through the end of the book, describe these events. Very little additional detail about this time is recorded in Scripture.
What will happen after God's plan of salvation has been completed and the earth purged by fire (II Peter 3:10-12; see "Does Peter Predict the Total Destruction of the Earth in II Peter 3:10?" for an explanation) is found in Revelation 21:
And I saw a new heaven and a new earth, for the first heaven and the first earth had passed away. Also there was no more sea. And I, John, saw the holy city, New Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband. And I heard a loud voice from heaven saying, "Behold, the tabernacle [dwelling] of God is with men, and He will dwell with them, and they shall be His people, and God Himself will be with them and be their God." (verses 1-3)
At that time, the earth will become the headquarters of God's Kingdom for the entire universe, and from that time forward it will never cease to expand (Luke 1:33; Isaiah 9:7). We can only guess what members of God's Family will do. Scripture provides us only a few clues.
God's purpose is concerned with all eternity. The sons admitted into His Kingdom will be active, dynamic, productive! They will not just sit around idly, but like the Father, they will plan and create. In a word, they will continue fulfilling God's master plan. Hebrews 2:6-8 says that God will ultimately put the whole universe—all things—under man's authority.
What is happening now is only the beginning. We cannot begin to fathom the immensity and glory of what the born-again sons of God will be doing for eternity. A great deal more is included than meets the eye in the hint given in Revelation 21:5: "Behold, I make all things new." Everything will be made glorious and wonderful beyond our most imaginative dreams (I Corinthians 2:9; Romans 8:18). God speed that day!