What the Holy Spirit teaches and reminds us of, is that God is love and this love is revealed by God the Father giving God the Son for the sake of the world.
One of the ways of speaking about the Holy Spirit, as always an inadequate way because we are using mere human words to speak of God, is to say that the Holy Spirit is the eternal love between the Father and the Son. Because the Father gives everything to the Son, including his divinity, the love he gives, the love the Son returns to him, is itself Divine, it is God the Holy Spirit.
Given to us, the Holy Spirit makes present in us the love of God which is God. Love, even human love cannot be taught in a lecture or in a book. Only by participating in love, by being loved can we learn to love. This is what the gift of the Holy Spirit does, it makes us participants in the love and truth and unity of God so that we can learn to love in unity and in truth. We receive this gift through Baptism and the Sacraments.
Another word for this is Sanctifying grace, it gives us the ability to hope to have faith to love, it gives our soul and thus us, new powers and a new way of living. Only under the influence of the Holy Spirit can we say Jesus is Lord.
The Holy Spirit in making present, making clear to us the love of God, can also be compared to light. We can’t really see light but without it we can see nothing. The story of the first great outpouring of the Holy Spirit at Pentecost indicates this. The listeners hear the gospel clearly and they understand it- they hear about the marvels of God and the resurrection of Jesus. It isn’t the Holy Spirit they hear but it is the power of the Holy Spirit that makes their hearing possible.