Old testament comes predominantly from Judaism.
Although the Hebrew Bible composed of the oral Torah as well as the written, the Old Testament is also part of the Torah. Basically, Old Testament is part of Judaisms core, but not the entirity of it.
The New Testament was a compiliation, I believe? The Gospels were written years after Christ had died, and then years later the Churches solidified the Bible which they thought composed accurate descriptions of Christ and geniune articles from the Gospels etc.
Christians usually don't take the Old Testament as seriously...If you've witnessed any Christianity-bashing, mostly anti-Christianity people just recite words and phrases from the Old Testament as an attempt to bash the Christian religion. Of course this is dismissed because Christians are mostly folllowers of the New testament.