First, let's get one thing clear: Jesus considered himself to be the one and only Son of God, as is clear from many other references.
Now, what of the incident in John 10?
Jesus said 'I and the Father are one'. His opponents reply,
"We are stoning you... for blasphemy, because you, a mere man, claim to be God."
Now what is often missed is that there are TWO ways that he could have defused this charge of blasphemy:
(1) By showing that he did not claim to be God; or
(2) By showing that it is not NECESSARILY blasphemy to claim to be God.
Which did he do? It was not (1) (as many people seem to suggest), but (2)!
Jesus answered them, "Is it not written in your Law, `I have said you are gods'? If he called them `gods,' to whom the word of God came - and the Scripture cannot be broken - what about the one whom the Father set apart as his very own and sent into the world?"
Surely the thrust of Jesus' reply is clear: if it's not blasphemy for part of their Law (Psalm 82) to call certain people 'gods', surely it is not blasphemy if applied to Jesus.
The obvious corollary is this: Jesus is more worthy to be called 'god' than any of the addressees of Psalm 82. Why is he greater? Because he called himself:
the one whom the Father set apart as his very own and sent into the world
And then in the next sentence, he put it even stronger:
I am God's Son
This passage is also notable for what Jesus did NOT say. If Jesus wanted to avoid the charge of blasphemy by denying he was God (defense (1) above), he could very easily have done so. But he did not. Rather, he showed (by quoting Psalm 82) that the claim to be God was not IN ITSELF blasphemous. His reply could be paraphrased, "It is not blasphemy to call someone 'god' if they deserve the title."
He then showed how he DID deserve the title, by describing himself as "the one whom the Father set apart as his very own and sent into the world", "God's Son", and by saying "the Father is in me, and I in the Father".
Jesus' opponents understood this claim clearly, so they tried to arrest him anyway (v.39). They followed his reasoning alright, they just didn't believe the claims he made about himself. Therefore they reasoned (correctly, if Jesus' claims were false), that Jesus was guilty of blasphemy.