What is the relationship that we should have with God the Father and the Holy Spirit? If the Holy Spirit is a person, should we pray to Him like we do to the Father and to the Son?
We do pray to the Holy Spirit.
O Heavenly King, the Comforter, the Spirit of Truth, Who art everywhere and fillest all things; Treasury of Blessings, and Giver of Life - come and abide in us, and cleanse us from every impurity, and save our souls, O Good One.
I was reading the Orthodox Church by Sergei Bulgakov and was bothered the he refers to the Holy Spirit as an "it" that is the love shared between the Father and the Son, rather than the third person of the Trinity. I understand now why he is considered a controvercial figure.
He also has a sub-Orthodox view of the role of the Saints in our salvation, asserting that God is not as approachable to us as the Saints, and thus we need to come under their cover. This seems to take the unknowability of God's Essence further than the Orthodox take it, and turns the Saints into Mediators, not merely Intercessors.