How are the persons of the Trinity distinctively active in a Christian's prayer? 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:


  “Again, truly I [Jesus] tell you that if two of you on earth agree
  about anything they ask for, it will be done for them by my Father in
  heaven" (Matthew 18:19 NIV)


We pray through Jesus. He is our Great High Priest.


  Therefore, since we have a great high priest who has ascended into
  heaven, Jesus the Son of God (Heb 4:14 NIV).


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. 


  We do not know what we ought to pray for, but the Spirit himself
  intercedes for us through wordless groans. (Romans 8:26 NIV)