Another great question from Sunday night, "My roomate is Jewish and therefore disregards the New Testament entirely. How can I explain the legitimacy and relevance of the New Testament?"
I think there are at least two possible approaches. First, to show from the Old Testament that the Messiah must suffer and die and be raised again. That is what Jesus did with his disciples as we see in Luke 24. Matthew and Lukes gospel will be helpful in this regard.
From the OT you can establish that man was created by God and hence he is our Creator King and deserves our full obedience (Genesis 1:26). Also, Genesis 3 shows that man did not live up to this duty/calling and falls under God's righteous judgment. Man is now separated from God and does not live in shalom peace with God, God's world or his fellow man. Mankind has gone far afield and is desperately wicked (Psalm 14:2-3). The Old Testament also establishes that life does not end with death. Mankind will be judged: some will enter into everlasting life and some to everlasting contempt (Dan 12:2). The OT also establishes that God is not only a God of justice and wrath, but of mercy and compassion (Ex 33:19, Ex 34:5-7). The Old Testament also established the need for forgiveness and the importance of satisfying the righteous wrath of God - that was the meaning of the Day of Atonement and all the OT sacrifices (the book of Hebrews summarizes, "Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins", Heb 9:22).
The Old Testament also shows that a better sacrifice would be needed to permanently atone for the sins of God's people. Isaiah 53 explains that a Suffering Servant would be appointed by God as this atoning sacrifice. Here are a few key verses from Isaiah 53:
"But he was wounded for our transgressions;
he was crushed for our iniquities;
upon him was the chastisement that brought us peace,
and with his stripes we are healed.
All we like sheep have gone astray;
we have turned—every one—to his own way;
and the Lord has laid on him
the iniquity of us all." (53:5-6)
"Yet it was the will of the Lord to crush him;
he has put him to grief;
when his soul makes an offering for guilt,
he shall see his offspring; he shall prolong his days;
the will of the Lord shall prosper in his hand.
Out of the anguish of his soul he shall see and be satisfied;
by his knowledge shall the righteous one, my servant,
make many to be accounted righteous,
and he shall bear their iniquities." (53:10-11)
Finally, regarding the OT witness, Psalm 16:10 is picked in the NT as a Messianic prediction that the Messiah would experience a resurrection (Acts 2 and Acts 13). This explains why Jesus, who was a Jew, was regarded as the Messiah by his followers, who were also Jews. The had met the risen Jesus and this event begged for an explanation. Jesus opened their eyes to see what was already there in the OT.
From a different angle, you can approach the validity of the New Testament using historical evidence for the resurrection. It is a historical fact that Jesus of Nazareth was crucified under Pontius Pilate. This fact is attested to by nonbiblical sources such as Josephus and Tacitus. It is a historical fact that his disciples preached that he had been raised from the dead on the third day. It is a historical fact that most of those disciples, all except Judas and John, gave their lives for his cause, refusing to renounce Jesus or go back to Judaism. It is a fact that the early church exploded in a context which was unfriendly to new religions. The new upstarts were persecuted by Jews and Romans. Yet, the church did grow and spread. Why? How do we explain the passion of these early Christians, their perseverance apart from the resurrection of Jesus?! How do you explain how people across the globe have come to worship a crucified criminal if something dramatic didn't happen, something like a resurrection? How do we explain the success of their message apart from an empty tomb? How would have believed Jesus was resurrected from the dead if his enemies could have produced a rotting corpse? The resurrection of Jesus, as crazy as it sounds (and it sounds crazy not only to us in the 21st century, but also to those in the 1st century, see Acts 17:32), is the most likely explanation for the historical message of the apostles and the success of their mission to the world.
If Jesus was raised from the dead, then there was something truly exceptional about him. If he walked out of the tomb on the third day as he said he would, he is a man to be reckoned with. What do you do with such a man? My suggestion, believe him! He is who he said he is. If he is who he said he is, we should listen to him and those who tell us about him.
There is a truck load of resources on this question. I would recommend taking a look at Two Ways to Live: Old Testament Version. It's a simple track that could prove useful. I would also recommend a book, though I haven't read it yet, by Christopher Wright, Knowing Jesus through the Old Testament. As I said, I haven't read it yet, but I have appreciated everything I have ever read by Chris Wright.
Again, hope that helps.