The narrative parts of the New Testament--the bits with story in them, i.e. the Gospels and Acts--were written 40-70 years after Jesus died. IOW, at least a generation later, after a lot of the original followers of Jesus had died. As best we can tell, the reason they weren't written earlier is because people assumed that Jesus would be coming back within their lifetimes, so there would be no need of passing things on to future generations. And then that didn't happen. There were probably some written accounts prior to those--collections of Jesus' teachings being the most likely--but a lot of what's there is an oral tradition. Nothing in the New Testament is attested to in the surviving non-Biblical records. Because the Romans did not care about the early Christians; they just saw them as a really weird sect of Jews. So they didn't write about them!
There are a couple of things that shaped what would come to be the Gospels and Acts that most Christians do not know about and that I think are really important if you want to a) understand Jesus and his first followers and b) start digging the antisemitism out of Christianity, or at least reducing it a little.
1) Jesus and his followers were all Jewish, and the split between Judaism and Christianity was long, uneven, and sometimes painful. Even after Jesus' resurrection, Jesus' followers worshiped in the Temple and their local synagogues. There was friction, but at the start nobody thought the Jesus-followers were starting their own separate religion. But a number of things happened to change this over the course of a couple of decades.
A) Paul/Saul. (He used both names for his entire life, depending on his audience--Saul among Jews and Paul among Gentiles.) Jesus' followers (including Saul/Paul) thought that Jesus was coming back soon. Once Saul became convinced that Jesus was the Messiah, he started seeking out pagans ... but he wanted them to not become Jewish. Why? Because there are several places in Hebrew Scriptures that talk about the Day of the Lord, and how "all nations" will come to Zion. IOW, people of all different ethnic/religious backgrounds will come to see the God of Israel as the best God out there. For Saul, having non-Jews who followed Jesus was proof that Jesus was the Messiah and that the Day of the Lord was here. So he wanted them to worship God, but not convert to Judaism. This caused a huge amount of friction both within the group of Jesus-followers, and also between Jesus-followers and the rest of the Jewish community.
B) As time went on, Jesus' followers went from "he's the messiah" to "he's the Son of God." Claiming he's the messiah was one thing; sure, if he were really the Messiah, why did he die and why aren't we living in a perfect Messianic Age? but still not a huge deal. There were a lot of would-be messiahs running around Judea in that era. But claiming he's the Son of God ... that's blasphemy and breaking the commandment of monotheism.
Together, these two factors (and probably others) eventually led to Jewish synagogues telling their Jesus followers that they couldn't be Jewish and a follower of Jesus; they had to pick. If you wanted to worship Jesus, you could not belong to your local synagogue. For every group, there comes a point where an offshoot or faction is so different from the original group that you can't really call them part of the same group any longer. This was deeply painful to the Jewish Christians who had to choose, and was apparently a fresh and major issue as the Gospels were being written.
Both John and Matthew have passages and story details that don't make sense if you're just looking at Jesus' life, but make a ton of sense if you realize that the people writing the Gospels are putting their own pain and anger at being thrown out of the synagogues back on the Jewish leadership that Jesus faced. Like the story of the man born blind in John 9-10. Jewish religious leaders weren't focused on rooting out people who followed Jesus during Jesus' own lifetime; we know this because a lot of Jewish religious leaders invited Jesus to their houses and considered him a fellow rabbi. But it was happening in the 70s and 80s.
This separation--and the pain and anger it caused--deeply shaped the way the stories in the Gospels and Acts are told. If you do not keep that in mind as you read these stories and think about them, you are going to swallow antisemitism without realizing it.
2) The second major thing that shaped the Gospels and Acts was the destruction of the Temple in 70CE and the mass enslavement and deportation of Jews from Judea that followed.
Both Judaism and early Christianity had the Temple in Jerusalem as their focal point. When the Romans razed it to the ground, both groups had to figure out "what do we do without the Temple and the sacrificial system?" Jews put the synagogues and the rabbis as the center of Jewish life. Christians responded by saying that Jesus' death was the fulfillment of the ancient sacrificial system, so it didn't matter any longer (cf. the Letter to the Hebrews in the New Testament). These are very different answers, which drove the two groups further apart and also started Christianity down the road of Supersessionism. Supersessionism, in a nutshell, is the idea that the Jews were only important to set up Jesus, and that Christianity has replaced them as the fulfillment of God's plan and as God's true people. So Jewish history and literature and religion only matter insofar as they point to Jesus and Christianity. I hope you can see how antisemitic this is?
As for the mass enslavement and deportations ... in the chaos of that, all the Jewish Christians who had maintained a Jewish identity died or disappeared. The only Jesus-followers left were the Gentiles. And they had no reason to like Jews and a lot of benefit to saying "yeah, we're Not Like Them, and we don't blame Rome for crucifying our God, we blame Them!" And that is what was going on when the Gospels and Acts were written. And it shaped how those stories were told.
Now, if Christianity had stayed a tiny weird mystery-cult, none of this would matter. But Christianity made a devil's-deal with the Roman Empire--give up some major theological beliefs (such as pacifism) in exchange for power. The Roman Empire already hated Jews because Jews refused to give up their culture and meld into the dominant Roman culture. And so Christianity took the most antisemitic possible interpretation of stories that were already hostile to Jews and made the worst possible readings the Standard Interpretations. And from that flowed centuries of murder, rape, torture, and theft.
And if you don't pay attention to this, you will blithely continue on in those antisemitic readings of the text.
If you are a Christian who wants to reduce the antisemitism in your Bible interpretations, go read Amy-Jill Levine's books. She's a Jewish New Testament scholar. The Misunderstood Jew: The Church and the Scandal of the Jewish Jesus is a good place to start. The Jewish Annotated New Testament is another good place to start.