Surprisingly, the name "New Testament" is a bad translation of Jeremiah 31:31-33, which says in the future God will make a "New Covenant" with his people.  The title of the New Testament should be "New Covenant."  The early Christians believed Jesus, his good news (gospel) and the Christians themselves, were a fulfillment of Jeremiah's prophecy.

The letters of Paul were written first, about 50 AD.  Next was likely the Gospel of Mark, written about 58 AD, by John Mark, Peter's missionary companion.  Peter's letters were written about 60 AD, to give encouragement and for theological reasons. Matthew was written about 65 AD.  James was written about 68 AD, by Jesus half-brother James (not the Apostle James).  Jude was written by another half-brother of Jesus.