One of the better examples, in my opinion, of how the New Testament sheds light on the Old, is the beginning of Matthew, the genealogy of Jesus. I used to just skip it so I could “get on with the good stuff”. Then it was pointed out to me that the purpose of the genealogy was this: Matthew was addressing educated Jews and numerology was very important to them. The genealogy sets forth 14 generations from Abraham to David, 14 generations from David to the Babylonian exile, and 14 generations from the Babylonian exile to the birth of Jesus. We have 3 sets of 14 or 6 sets of 7. Jesus was the seventh seven, representing perfection (the number 7 represents perfection). This has nothing to do with prophecy, as far as I know, but it is a truth, and an astounding truth at that, just based upon the facts of the family tree found in the Old Testament. Did it just turn out that way? Maybe. Or does it mean that which Matthew implied it meant?
As Christians, as Anna said, we really cannot separate our knowledge of the New Testament from the Old Testatment. A Jew must read the Old Testament with the assumption that Jesus is not the person to which it is pointing. Yes, it is somewhat of a conundrum and I don’t see any way around it.
As far as the realtionship between Jews and Christians we can never forget that the Jews are our elder brothers and sisters in the faith and God’s chosen people. They have never stopped being God’s chosen people. We believe it has been fulfilled in Jesus Christ, they do not, but Judaism is intrinsic to Christianity, We even have a Jew as the head of our Church.