Now, to answer your question, "Do you believe when the Rapture comes that the Holy Spirit will go with us or stay here on earth?" Stay with me here, and I will get to it.
There are three views on the Book of Revelation and end times. The first is your view, pre-millennialism. Another is post-millennialism, and another is "a-millennism." Let me explain them. I graduated from New Orleans Baptist Theological Seminary in 1976 and did a Master's thesis on Revelation.
Post-millennialism believes that the world will eventually receive Christ and that a Christian world will welcome Christ into His kingdom. That great missionary hymn, "We've a Story to Tell to the Nations" is written by a post-millennialist. You can look it up online and get a feel for this view. This is the chorus:
For the darkness shall turn to dawning,
And the dawning to noonday bright;
And Christís great kingdom shall come on earth,
The kingdom of love and light.
It was written in 1896, the height of post-millennialism. Most Southern Baptists in the late 1800's and early 1900's were post-millennialists including Dr, B. H. Carroll, president of Southwestern Baptist Theological Seminary in Ft. Worth.
This view faded from popularity with the outbreak of World War I when it seemed Christ's great kingdom would not come on earth.
It was replaced by your view, the pre-millennial view. The man who popularized this view was a lawyer, C. I. Scofield, who had the nerve to put his notes into the text of the Bible. It was published in 1917, during the heat of World War I. His Bible is still popular today and is called the Scofield Bible. I feel it is rather audacious to put human notes into the context of the Bible because many will believe that the notes are as inspired and infallible as the Bible. That's what happened with Scofield's notes. People began believing his notes were the Bible since they were incorporated into the Bible.
Pre-millennialists believe that the world will grow worse and worse. Christians will become less and less. Then, when the world is at its worse, Christ comes again to rapture his church out. While Christianity is under attack in many countries - like the USA - the faith is growing rapidly in other places like in Africa and South Korea. For every person born in S. Korea, three are being born again. We Americans often fail to see the big picture of world evangelization and the progress and advance of Christ's kingdom.
Pre-millennialists take the Book of Revelation literally. For example, the 1000 years (the millennial) is a literal 1000 years. In my study, I concluded that Revelation is symbolic. For example, in chapter 1, John has a magnificent vision of the ascended Christ. His eyes are fire. His feet are bronze, and his head and hair are white like wool. All of these are descriptive symbols of Christ and all are loaded with meaning. Now, if chapter 1 is symbolic, then the rest of the book is too. If not, then the Jehovah Witnesses are correct in their teaching that only 166,000 will make it into heaven. The pre-millennialists then take some things literal in Revelation and some things symbolical. In my opinion, it can't be both. Interpretation must be consistent all the way through.
Every number in Revelation is loaded with symbolism and meaning. For example, 666 is the number for incompleteness. 7 or multiplies of seven like 144,000 are the number for absolute perfection. Pre-millennialists believe that 666 will actualy be stamped on the forehead of those marked with the number of the beast. I believe that those who are without Christ are marked by the devil - just look at them and it is plain to see as any number stamped on their forehead.
Another issue is Paul's description in 1 Thessalonians 4:15-17. According to the Lord's own word, we tell you that we who are still alive, who are left till the coming of the Lord, will certainly not precede those who have fallen asleep. 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. 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.
Again, this passage is loaded with symbols. Our problem in part is that we are influenced by the Greek culture which takes everything literal and concrete. The Hebrews, on the other hand, wrote in symbols. The majesty of spritual truths was simply too great to be able to put into words, so they wrote in symbols using words to make the symbols.
I believe that if you die today, and I am alive tomorrow and Jesus comes tomorrow, that you will have risen first. I then would be taken alive to be with him tomorrow. The problem here is with time. Do the tombs open, and then BOOM, Christ comes back. The rapture as it is called in pre-millennialism.
This view holds that when we die, our soul goes to be the with the Lord to float around in heaven in a disembodied state waiting until the rapture when our soul is reunited with our body thus forming a spiritual body that is recognizable by others in heaven.
When I study the Mount of Transfiguration (Matthew 17), Moses and Elijah who had been dead for over a thousand years, were fully recognizable and in their spiritual bodies when they appeared with Christ and appeared to Peter, James, and John. They were not disembodied souls floating around. They stood on the earth.
You see, the Greeks believed that we are a body, a soul, and a spirit - three separate parts, and pre-millennialism has adopted the Greek view - the soul goes to heaven and the body is left behind. The Hebrews believe that we are soul, body, and spirit - inseparable - like the Trinity. Christ is one with the Father and the Spirit.
I firmly believe what Paul said, "T be absent from the body is to be present with the Lord" (2 Corinthians 5:8).
The pre-millennialists have Christ coming back with the souls of his saints. Then, the graves open and the body and the soul are re-united and given a spiritual body as the physical body is united with the soul in the air. That's the first second coming, and then he has a second second coming when he comes back again to bind and throw the devil into hell. Jesus spoke of only one second coming.
Now to "a-millennialism." This view has been the historic position of the church. It means "no millenniam." Paul himself looked for Jesus to return befored he died. Peter referred to the "Last Days" as the days they were currently living in (Acts 2:47). "These are the last days the prophet Joel spoke of." And in Hebrews 1:2 "but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom he made the universe." The New Testament teaches that the Last Days began with the resurrection of Christ and not some time in the future when all the "signs" would be fulfilled.
The a-millennial view holds that Revelation and other New Testament referrances about Last Things are symbolic. Even our Lord spoke in symbols. "I am the vine" is a sympol of great eternal truth and not to be taken literally that He is a vine growing in a vineyard.
So with all that said, let me answer your question. "Do you believe when the Rapture comes that the Holy Spirit will go with us or stay here on earth?" If I held to your pre-millennial view, I would say that the Holy Spirit stays on earth because God is omnipresent meaning that He is everywhere. There is no place where God is not present. Now, God, the Father was not presnt with Jesus at the moment He died. "My God, My God, why have you forsaken me." And, I don't think God is in hell. By His power, he created hell and sustains hell as a place for eternal punishment, but I think those people are fosaken by God, and He is not there. I don't think or know of anything in the Bible where His Spirit ever leaves this earth. Paul affimed, For by him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things were created by him and for him. He is before all things, and in him all things hold together (Colossians 1:16-17). If the Spirit left this earth, I don't think it would hold together according to this Scripture.
Know that some of my pastor friends are pre-millennialists and we have some lively, friendly debates about these wonderful doctrines. But, whatever view we hold, we know one thing. We are all pan-millennialists - this will all pan out the way God wants it too!