Another way of asking this question is by saying, "When God interacts with praying Christians, how do the persons of the Trinity interact in ways that neither of the other two persons do?" I'm asking from a biblical, Protestant perspective.
My understanding from growing up in an Anglican church:
We pray to God the Father. He is the one that answers our prayers. Not a perfect verse for this, but the only one I could think of:
We pray through Jesus. He is our Great High Priest.
Think back to the old testament, the High Priest was the one who could go into the holy of holies in the temple and gave access for the people to talk to God.
The Holy Spirit prompts us to pray, and directs us to pray for certain things - e.g.