What's the main reason(s) Jews don't accept what Christians call the New Testament? . .
As others have mentioned, that is a very complex question, which involves many reasons.
But the nickel version of the answer is that, first of all, we don't believe that Jesus was the messiah (to be the messiah in Judaism means something very different than in Christianity), nor do we believe he was any kind of prophet.
Second of all, we believe that the One God is entirely, completely, and without exception, One. Indivisible. We also believe that He is entirely without any kind of physical form, nor is any physical form capable of containing HIm. Which means that we cannot believe that He could appear in the form of a man, with a physical body.
Third of all, we do not believe in the concept of Original Sin. Therefore, there is nothing for which humanity requires a savior such a Jesus is held to be by Christians.
Fourth, it is entirely incompatible with Jewish belief to suppose that anyone could institute a new covenant to supplant the covenant of Sinai. The only way such a thing could happen would be if God appeared again to all the Jewish People at once (not in the form of a human being, remember, because we don't believe such a thing is possible), just as at Sinai, proposed a new covenant, and the entirety of the Jewish People simultaneously agreed to it and accepted it.
Fifth, the Torah and the rest of Jewish Law make clear that if a person comes around and claims to be a prophet who speaks for God, and then goes on to tell us that God wants us to stop observing the commandments and cease obeying the precepts of Jewish Law, that is itself a sign that the individual is a false prophet who does not speak for God; because God would never, ever tell us that.
Sixth, Jewish tradition and sacred text (such as the Tanakh, or "Old Testament") is not intended for non-Jews. It was written by Jews, for Jews, in the Jewish language, for use with Jewish methods of interpretation and understanding. We do not recognize that anything in it pertains to Jesus, or has anything to do with Christianity.
That's the very short version, but I'm sure you can see some of the fundamental issues.