I get the impression that the Bible says that Jesus the Son existed eternally before the universe was created and that he was omniscient. What about when he was born on Earth? Did he still know what he knew before? Or did he only have access to that knowledge at a later point in time?
The answer to this question leads to a confessional paradox. That is, it's a paradox and yet it ought to be confessed as true even if it doesn't necessarily make complete sense.
Jesus, because He's God, knows all things.
Jesus, because He's human, says there are things He doesn't know (see where Jesus says that of the day and hour of His return only the Father knows, not the angels, not the Son, but the Father only).
We read that Jesus "grew in wisdom before God and man". So Jesus, a human being, went through infancy and childhood and into adolescence and finally adulthood--He grew in wisdom, He learned things. We can deduce that Joseph and Mary were there teaching Jesus how to speak, encouraged Him to crawl and walk, they had to potty train Him. He would have been educated in some fashion through the regular synagogue life of the Jewish community in Nazareth.
We also see that when Jesus was an adolescent He was at the Temple having deep religious conversations with religious experts.
What need does God have to learn when He knows all already?
How can the All-Knowing not know something?
How can Jesus both know everything, and not know somethings or have to learn things?
That's the paradox. The Incarnation introduces many paradoxes.
God can't die. Yet God died on a cross.
God can't suffer. Yet God suffered, He was beaten and cursed, His beard yanked, had a wreath of thorns forced upon His head.
The easy way out is to try and separate Jesus' Divinity from His humanity (or vice versa). But for the Incarnation to truly mean something we must confess the true and indivisible unity of Jesus' singular Person. We don't have a Divine Person on the one hand and a human person on the other. We have a single Person: God and human. So we speak of the Person growing, the Person knowing, the Person suffering, etc. Jesus grew in wisdom: God grew in wisdom. Jesus commanded the waters of the sea to be still: a man commanded nature.
When Christians talk about "The Mystery of the Incarnation" it's not "mystery" in the sense of "Well, we just don't know anything"; it's that we are confessing and believing something revealed to be true; even when it is immensely difficult. That's what a Mystery is: truth revealed outside of human reason.