I also want to give credit to Tim Chester and his book "Delighting in the Trinity." He does a really good job of covering the history of how the Church has developed its view of the Trinity, and he writes in such a way that reveals to the reader how important the doctrine of the Trinity really is. I would recommend anyone wanting to get an introductory look at the doctrine of the Trinity to check out this book.
Chester, in my opinion, summarily nails the deal about salvation when he writes:
Christ's death of the cross can be seen as His victory over Satan and as a moral example. These approaches see salvation as a transaction between God and Satan and between God and humanity. But at its heart salvation is a transaction within the Trinity. The Son offers Himself to the Father as our substitute. God both judges and is judged. And the Spirit applies this transaction to our lives.The Godhead: the Father, the Son, and the Holy Spirit, work in union. It is the Father that gives the world His Son. The Son who gives us His Holy Spirit. And in reverse, it is the Holy Spirit that reveals God's Word, Jesus Christ, and it Jesus Christ who says that he who knows Him also knows the Father. What incredible stuff (for lack of a better theological word)! It is God and God alone in the Trinity that works out our salvation!
Without the Holy Spirit, we would not know Christ. Without Christ and His work on the cross, we could not be reconciled to the Father and be in relationship with Him. And it is God, His Word, and the Spirit hovering the waters in the middle of creation. The doctrine of the Trinity is paramount. The relationship between the atonement and the Trinity operates in both directions. The Trinity is the foundation of the atonement and the atonement is the ultimate revelation of God's trinitarian character.
It is God that saves us. Not us. In His love, He sent His Son to reconcile the all humanity to the Father. It is the Holy Spirit, whom the Son sent, that quickens and gives us faith at the hearing of His Word. It is God, as the Trinity, that provides salvation.
Almighty and everlasting God, who has given to us, Your servants, grace, by the confession of a true faith, to acknowledge the glory of the eternal Trinity and in the power of the Divine Majesty to worship the Unity, we implore You that You would keep us steadfast in this faith and evermore defend us from all adversities; who lives and reigns, Father, Son, and Holy Spirit, one God, now and forever. (Collect for Holy Trinity)