One of the greatest mysteries of the Bible is the Trinity. We believe that God has revealed Himself as Father, Son, and Holy Spirit, which are not separate gods. Each member of the Trinity has a unique role yet they are one. They also play a part when it comes to the salvation of souls.
Below is how Graham Ryken and Michael LeFabvre describe each member of the Trinity's role in salvation:
The Father: Salvation originated with the Father. Ephesians 1:3-6 tells how the Father chose us before the foundation of the world, and predetermined our adoption as this children through Jesus Christ. The Father is the administrator of salvation, and he oversees the process from beginning to end.
The Son: Salvation is brought to fruition in the Son. Everything the Father does for our salvation, he does through Christ. The work of the Son means redemption, adoption to the Father, reconciliation, sanctification, and glorification (Ephesians 1:7-12). It operates horizontally as well as vertically, and it is for Jew and Gentile alike. It is through the Son that we achieve salvation and come into full relationship with the triune God.
The Spirit: Salvation is communicated by the Holy Spirit. The Spirit changes us from the inside out, preforming the gracious act of regeneration. With this comes the gift of faith and the spiritual ability to believe in the Resurrection. Through the Holy Spirit, our salvation becomes a present reality, applicable to our lives in our own specific context. It is the work of the Holy Spirit in our lives that serves as a seal, establishing us as children of God (Ephesians 1:13-14).
Source: The Work of the Trinity in Salvation which is adapted from Ryken and LeFabvre's book, Our Triune God: Living in the Love of the Three-In-One
Recommened Books on the Trinity:
The Deep Things of God: How the Trinity Changes Everything by Fred Sanders
Holy Trinity: In Scripture, History, Theology, and Worship by Robert Letham
Delighting in the Trinity: Why Father, Son and Spirit are Good News by Tim Chester
The Forgotten Trinity by James White