I would tell you that although it is no small task, a careful read of the Talmud would show you a very Jewish Jesus. It would also show you that many of his teachings were not as new as many today suppose them to have been. He frequently threw rabbinical law in the faces of the Jewish elite, and many of his known conversations were shown to have been tests. The various groups, for example, each tested his learning and doctrine. One asked him about the greatest commandment. The Sadducees tried to challenge his very Pharisaical point of view on a resurrection. Reading through the principles discussed in the Talmud, you can even see Jewish wisdom in practice. By example, the Jews believed that the wise man didn't have the best answer, but the best question, and you can usually see Jesus answering baited questions with rhetorical questions to establish the truth of the matter. Whose image and superscription is on the coin? Which of you would not pull his ox out of a ditch on the Sabbath?
So the New Testament evidence would show that Jesus was a solid student of Jewish wisdom and learning. He was, in effect, a good Jew. But his talk of the kingdom was also radical talk, and dangerous. Talking of the kingdom of God was not too far different then than Muslims talking about Jihad today. The Jews were a ceaselessly rebellious people. One of Jesus' own followers was a zealot, no doubt because Jesus' message struck cords with the discontent due to his kingdom message.
The real and substantial reason for his fame is the resurrection. His claims were proved when God raised him from the dead. But this truth made the Jewish authority guilty of murdering God's messenger, son, etc. (however it is that they chose to perceive it), and so rather than being important to them, he became a threat.
There's no substitute for education. If you want to be able to defend your faith against an unbeliever, I would strongly advise that besides studying your Bible carefully, you also get a copy of Josephus (to learn about first century Jewish culture from a first-hand source), and I would suggest reading the Talmud (to get a firm grasp on the intricacies of Jewish law as it existed during that time period). Apart from a few juicy tidbits, you won't really get enough information on a web forum to adequately defend your faith against an atheist professor with a college education in theology.