You're asking a lot of different questions here so I'll do my best to provide as helpful an answer as I can.

First, the question of why the disciples of Jesus became so interested in spreading his teachings. That behaviour is generally attributed to something that has become known as the Great Commission. A version of the event exists in some form in Mark, Luke, and John, but the most familiar account comes from Matthew 28:16-20 (NIV):

Then the eleven disciples went to Galilee to the mountain where Jesus had told them to go. When they saw him, they worshiped him; but some doubted. Then Jesus came to them and said, "All authority in heaven and on earth has been given to me. Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age."

As for the success of early Christianity, a great deal of credit must be given to the message. As you can see quite clearly from examples like the Beatitudes (Matt. 5:3-12, Luke 6:20-22), the tenets of Christianity were heavily directed towards the masses. We see the message being focused on the meek, the hungry, and the persecuted and that contributed a great deal to the mass appeal of Christianity in the early days.

It must be noted though that the term 'early Christianity' is, at best, a catch-all. The interpretations of the teachings of Jesus had as many different varieties as there were people to interpret them. From the second century onwards there were considerable debates between Christians who couldn't agree on aspects, and many couldn't even agree on the nature of Christ: some said he was mortal, some that he was purely divine, and some that he was mortal and then made divine at baptism. It was for these reasons that the various Ecumenical Councils were convened - to set the record straight. They weren't even necessarily all that effective in that respect: those who disagreed would simply break off and make their own church. As an example: the Council of Chalcedon (451 CE) said that Christ was both divine and human. Those who rejected it made their own church in Alexandria.

Much of what has survived today has come down from the writings of Paul in the form of the Pauline epistles which make up almost half of the New Testament, so it is his views that have been carried along as the typical Christian viewpoint. But these works were compiled into the current Bible by the Ecumenical Councils so it is difficult to not see Christianity as one coherent and uniform movement.

I seem to have gone a bit outside of the range of your question, but I'll sum it up here. According to the Gospels, the Great Commission was the set of instructions that proclaimed their authority to spread Christianity throughout the world. The disciples went off and began their conversions by appealing to the masses. Why they were so receptive to the new message could be a variety of reasons but I can admit I'm not as knowledgeable in those theories. What emerged was a Christianity with many different centers and each with different conceptions and different understanding of the ancient writings (of which there were more than survive today). Eventually the Ecumenical Councils were convened to 'set the record straight' and what survives today in the form of the many different denominations and movements got their start from them.

I hope I was able to clear things up a little for you, or at least gave you some more questions to ask. Much of my information comes from Bart Ehrman's The New Testament: A Historical Introduction to the Early Christian Writings, 2008. He has since published a more recent edition (2011) so if you discover that anything I've said has since changed please let me know. Otherwise there are scores of sources by C. H. Dodd and Ehrman that are worth looking into.