The saints will reign on the earth as stated in Revelation 5:9-10 (NKJV):
9 And they sang a new song, saying:
“You are worthy to take the scroll,
And to open its seals;
For You were slain,
And have redeemed us to God by Your blood
Out of every tribe and tongue and people and nation,
10 And have made us kings and priests to our God;
And we shall reign on the earth.”
I am aware of textual variations here, but the sense of the earth being what is reigned over is the same.
Daniel 7 also makes it clear that the earth is the kingdom that the saints of the Most High receive:
17 ‘Those great beasts, which are four, are four kings which arise out of the earth. 18 But the saints of the Most High shall receive the kingdom, and possess the kingdom forever, even forever and ever.’
23 ...‘The fourth beast shall be
A fourth kingdom on earth,
Which shall be different from all other kingdoms,
And shall devour the whole earth,
Trample it and break it in pieces.
26 ‘But the court shall be seated,
And they shall take away his dominion,
To consume and destroy it forever.
27 Then the kingdom and dominion,
And the greatness of the kingdoms under the whole heaven,
Shall be given to the people, the saints of the Most High.
His kingdom is an everlasting kingdom,
And all dominions shall serve and obey Him.’ (NKJV)
Verse 6 you quoted in your question concerns the 1000 year reign with Christ:
5 But the rest of the dead did not live again until the thousand years were finished. This is the first resurrection. 6 Blessed and holy is he who has part in the first resurrection. Over such the second death has no power, but they shall be priests of God and of Christ, and shall reign with Him a thousand years. -Revelation 20:5-6 (NKJV)
This particular time span is when the saints are with Jesus in heaven. This first resurrection is when Jesus returns to gather His saints:
39 "This is the will of the Father who sent Me [Jesus], that of all He has given Me I should lose nothing, but should raise it up at the last day. 40 And this is the will of Him who sent Me, that everyone who sees the Son and believes in Him may have everlasting life; and I will raise him up at the last day."
44 "No one can come to Me [Jesus] unless the Father who sent Me draws him; and I will raise him up at the last day." -John 6:39-40, 44 (NKJV)
And after He resurrects them, He will take them to the mansions prepared for them in His Father's house:
1 “Let not your heart be troubled; you believe in God, believe also in Me. 2 In My Father’s house are many mansions; if it were not so, I would have told you. I go to prepare a place for you. 3 And if I go and prepare a place for you, I will come again and receive you to Myself; that where I am, there you may be also." -John 14:1-3 (NKJV)
Paul speaks of this as well in his letter to the Thessalonians:
15 For this we say to you by the word of the Lord, that we who are alive and remain until the coming of the Lord will by no means precede those who are asleep. 16 For the Lord Himself will descend from heaven with a shout, with the voice of an archangel, and with the trumpet of God. And the dead in Christ will rise first. 17 Then we who are alive and remain shall be caught up together with them in the clouds to meet the Lord in the air. And thus we shall always be with the Lord. -1 Thessalonians 4:15-17 (NKJV)
So as you can see, there is a period of time where the saints are received to Jesus who takes them to the place He has prepared for them in His Father's house. This is in heaven, for heaven is His throne, and the earth is His footstool. This is the first resurrection where the resurrected saints reign with Jesus in heaven 1000 years.
The earth at this time is melted away with fire:
10 But the day of the Lord will come as a thief in the night, in which the heavens will pass away with a great noise, and the elements will melt with fervent heat; both the earth and the works that are in it will be burned up. 11 Therefore, since all these things will be dissolved, what manner of persons ought you to be in holy conduct and godliness, 12 looking for and hastening the coming of the day of God, because of which the heavens will be dissolved, being on fire, and the elements will melt with fervent heat? 13 Nevertheless we, according to His promise, look for new heavens and a new earth in which righteousness dwells. -2 Peter 3:10-13 (NKJV)
The earth is not melted away instantaneously, for all the birds have time to gorge themselves on the corpses of those who received the mark of the beast:
19 And I saw the beast, the kings of the earth, and their armies, gathered together to make war against Him who sat on the horse and against His army. 20 Then the beast was captured, and with him the false prophet who worked signs in his presence, by which he deceived those who received the mark of the beast and those who worshiped his image. These two were cast alive into the lake of fire burning with brimstone. 21 And the rest were killed with the sword which proceeded from the mouth of Him who sat on the horse. And all the birds were filled with their flesh. -Revelation 19:19-21 (NKJV)
The saints will then take part in the Judgment of the world and angels, of which Paul briefly alludes to while rebuking the Corinthians for suing each other:
2 Do you not know that the saints will judge the world? And if the world will be judged by you, are you unworthy to judge the smallest matters? 3 Do you not know that we shall judge angels? How much more, things that pertain to this life? -1 Corinthians 6:2-3 (NKJV)
Revelation 20:4 also mentions this partaking in the Judgment:
And I saw thrones, and they sat on them, and judgment was committed to them.... -Revelation 20:4 (NKJV)
As does Jesus in Matthew 20:
27 Then Peter answered and said to Him, “See, we have left all and followed You. Therefore what shall we have?”
28 So Jesus said to them, “Assuredly I say to you, that in the regeneration, when the Son of Man sits on the throne of His glory, you who have followed Me will also sit on twelve thrones, judging the twelve tribes of Israel. -Matthew 19:27-38 (NKJV)
After the 1000 years, the second resurrection takes place. While the phrase "second resurrection" is not there, the concept is definitely portrayed when it says
But the rest of the dead did not live again until the thousand years were finished.... -Revelation 20:5 (NKJV)
The devil is then thrown into the lake of fire and the judgments that were decided during the 1000 years are executed, with all those not being found in the Book of Life being cast into the lake of fire (v15), which is the second death (Rev 21:8).
God then creates the earth anew:
Now 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. -Revelation 21:1 (NKJV)
This new earth has no more death, sorrow, crying, or pain, "for the former things have passed away" (v4, NKJV).
- The saints reign with Jesus in heaven for 1000 years, taking part in the Judgment of the world and angels.
- After the 1000 years, the judgments are executed and the earth is remade anew.
- The saints then reign with Jesus on the newly remade earth. While we are not told what exactly will be on the remade earth other than the New Jerusalem, we are told what will not be there: death, sorrow, crying, and pain.
- Adam and Eve had dominion over the earth before the Fall while there were no other humans. They reigned over the earth, which included all plant and animal life. All things will be created anew, which would also include plants and animals. The saints will rule over everything in the newly remade earth.