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)
In the Anglican Communion the most common idea is that what one personality knows the other knows (you should rather use personality than the word person when talking of the Trinity in Unity.) If you pray to The Father, the Son and The Holy Spirit hear you also. We address our prayer to The Father because Jesus taught His disciples to pray to The Father. We open with "Our Father," rather than just "Father." LDS Priests usually address the prayer to "Heavenly Father." Anglican Priests often address their prayer to "Father Almighty," and often to "Almighty Father." which is a different way of thinking of God. In RC doctrine, prayers are often addressed to Mary (theotokos) asking her to intervene with the Son, on a persons behalf (this was recognized from the experience at the Wedding in Cana.)
There are so many millions of people who have been helped by prayers to Jesus, to Saints, to Mary, to Joseph, to Yahweh, to Mohammet, that it is evident God hears every prayer that the Spirit recognizes has God as its ultimate destination.