There are no parts to God that build to a whole. Each portion of the trinity is fully God, just a different manifestation. The Father is the God spoken of throughout the old and new testaments. Jesus is God in human form sent to Earth to teach and then to be offered as the ultimate, final sacrifice for the sins of mankind. Jewish law required blood to be shed for sin, as the "wages of sin" were "death." Usually it was a lamb or goat or bull substituted for the sins of men, which is why Jesus is also referred to as the "Lamb of God." And the Holy Spirit is the manisfestation of God that works within the lives of his followers throughout the Bible but most specifically in new testament times through the present. Like the Wiki article pointed out, the concept of the Trinity is to explain to men what they could not possibly comprehend, and the fish analogy seemed appropriate. Even the fact that Jesus took on a physical representation can still be baffling, as you pointed out in your question of why did he pray. You could take two (or more) views on that. 1) To teach. He had to show his disciples how to speak to God, what to ask for, what to desire, etc. 2) While fully God, Jesus was also man. During times of prayer, it was his humanity seeking the divine presence of God.