Jesus and the Apostles spoke Aramaic, not Hebrew. Hebrew was a language of scholarship, like Latin later on in the West. Jesus was of course fluent in Hebrew but that was not the language he spoke to people in.
The nature of the religion that Jesus founded, as opposed to Paul of Tarsus, may be seen in the community of pre-Pauline Christians (or Nazarenes) in Jerusalem, which was headed by the Apostles after Jesus was crucified. They clearly saw themselves as a Jewish sect, or as the true fulfilled Judaism. Considering that the Apostles knew Jesus intimately, while Paul's early encounters with his followers mainly took the form of him trying to kill and imprison them, I think it's safe to say that the Apostles understood Jesus' original inclination rather better than Paul did.
We may see four main phases in the evolution of Christianity since the Crucifixion. First was the brief period of Jewish Christianity, in which the followers of Jesus continued to follow his teachings under the direction of the Apostles, mainly Peter and James, in a community centered in Jerusalem. This included following the Mosaic law and Jewish religious practices, as modified by Jesus' teachings.
The second phase began when Christianity became a religion of the gentiles, separate from Judaism, and Paul of Tarsus was more instrumental in that happening than any other person. His ministry started a flowering of diverse Christian practice throughout the Roman Empire, due to the fact that it was technically illegal but the prohibition of it was seldom enforced; what this meant in practice was that Christians could practice their faith without fear, but also without any government support or enforcement of orthodoxy.
The third phase, the Imperial Church, began with the Council of Nicaea. Christianity became first a legal religion, then the state religion of the Empire, then the only permitted religion, and the power of the state supported it and greatly narrowed diversity. The Roman Catholic and Eastern Orthodox Churches both descend from the Imperial Church, the split between them occurring shortly after the fall of the Western Empire.
The fourth phase began with the Protestant Reformation. It has seen the shattering of the Church's political authority and its ability to suppress heresy, and thus a return to diversity and freethinking.
The books of the canonical New Testament, along with others, were mostly written during the second phase, but the choice of which texts are canonical was made in the transition to the third, and so the New Testament is an artifact of the Imperial Church. If you really want to understand the spirit of pre-Imperial Christianity, take all of those books and mix in every non-canonical Christian writing from the period you can find (the Gnostic texts, the Gospel of Thomas, non-authenticated apostolic letters, etc. -- everything), and don't make a distinction among them. Here are a couple of sources:
NonCanonical Gospels and Other Writings
Gnostic Society Library: Sources on Gnosticism and Gnosis
