exactly how does this spirit of God work?

Answer 1

God who is spirit has a spirit which is called the Holy Spirit. The Holy Spirit is the creative force and power which comes from God to bring into existence what ever He speaks. When you go back to Genesis 2 it says the Spirit of God was hovering over the water waiting for God to speak. Likewise, when we as believers are walking in God's will we can speak God's Word and the Holy Spirit will move for us and bring into existence whatever we speak. As long as it is in God's will. It may take a while for it to come to pass, but the Holy Spirit will line up the events and circumstances for whatever you prayed for to come to pass.

Answer 2

There was lots of discussion about this in the 350s and 60s AD.
The church was trying to make sense of the experience they had of knowing something of God's presence with them and in their lives. Sometimes this was in a very physical way - like the tongues of fire that landed on the heads of the people in Acts chapter 2 - other times it was in terms of the Spiritual Gifts (supernatural healing, prophecy, hospitality, a prayer language called "speaking in tongues" etc) or Spiritual Fruit (dramatic increase in love, joy, peace, patience, etc in their lives) as they continued to worship God.
This seemed to tie in with what Jesus promised about him sending an encourager who would make sense of what he was saying to them and guide them into all truth etc. (John's gospel talks a lot about this but they all mention it)

A group called the Capadocian Fathers argued that just as they had decided that Jesus must be divine, that there was a spirit of God that was a distict person in his own right and that he was also Divine (so the idea of One God who is three persons).

The same methods work today in our understanding of the Holy Spirit (or pneumatology) - the spirit seems to work independantly of our models of what he will do so we have to keep forming our models out of our experience and observation of what he does rather than say "this is what he does" and refusing to acknowledge him doing anything outside this.
Most Christians experience some of the spiritual gifts and fruit that I mentioned earlier (though God is able to give or not give as he sees fit and in 1 Corinthians Paul is pretty harsh in saying that the presence of spiritual gifts doesn't make you more spiritual or filled with the spirit).
The Holy Spirit is given to Christians (often at the point of conversion) it may come in an obvious way (sometimes including things like falling over or a sense of electricity or warmth - as Charlse Wesley did) or it might be nothing - God can do it as he wants.

Christians can ask GOd for more of his spirit in their lives (or more of their lives given over to the Spirit) and many in the charismatic and pentecostal wings of the church will pray "Come Holy Spirit" (as the early church does in hte Bible) - The Eastern Orthodox church also has a very strong theology of the Holy Spirit - believing in "theosis" that we can be increasingly joined to God in this life through his spirit. The western church has had lots of strands of this too (eg many of the the mystics in the middle ages - such as Bernard of Clairvaux or St John of the Cross) but most of the leading thinkers of the evangelical tradition also experienced something along these lines (eg Luther understood union with God in this life through meditation on the Bible and the sacriments) -

If you're interested in encountering more of the holy spirit yourself I'd recomend this thinking about scripture. Perhaps read John's Gospel asking God to help you to open up to more of his spirit...

Hope this helps!

Grace x