Although the teachings at the heart of Christianity owe their birth geographically to the Middle East, specifically Jerusalem and the surrounding environs, as an organisation it really has Rome to thank for its initial development. Peter, often regarded as the natural successor to Jesus established the Roman connection but it is Paul we have to thank for the spread of what today is regarded as the Christian faith. Modern Christianity is often termed "Pauline" Christianity as a way of indicating that the message is that of St Paul rather than directly of Jesus himself. In fact scholars have made much of the differences of the original message of the Jerusalem church that headed by Jesus and James and the evolution of ideas that came about through Paul's teaching. It has to be borne in mind that Paul was a Roman citizen and probably never met Jesus personally and his brand of enlightened Christianity found a favour in the Roman Empire that the more radical teachings of Jesus would have.
So Paul's rebranded Christianity grew steadily through out the Empire but it was to have its base in Rome. Initially it was a very secretive organisation, for good reason. Roams were obliged to acknowledge that their emperor was in fact a living god. It was bad enough that the pagan ideas of the empire promoted a pantheon of hundreds of gods but to suggest a mere mortal was a god was too much. If they were seen to actively oppose this idea it would have been the end of their movement so the best way to deal with the problem was to worship in secret, private rooms and followers' houses were used instead of public gatherings. The idea of Christians being thrown to the lions is something that has been taken to an extreme, there are a few recorded incidents of Christians being put into the arena, but the idea of wholesale religious persecution is a false one. Roman religious policy was generally very accommodating as long as people were seen to be following the laws in public. The odd sacrifice would do just to cover look the part in public, what people got up to behind closed doors was their own business.
In the third century a number of factors came into play that paved the way for the eventual dominance of Christianity. In 274 the emperor Aurelian believed that the sun god had brought him victory in his campaigns in Syria. He set up a state cult to Sol Invictus, the unconquered sun: and announced that the birthday of the sun, a day of special festivity, was December 25th. Christianity would, later borrow the suns halo, as well as the date of the birthday. Later, Diocletian declared that Mithras, who was very popular with the army, to be the god who was the "protector of the empire" After slaying a bull, Mithras made a sacrificial meal of it: and his followers re-enacted this with a ritual of bread and wine. Although the Christians were outraged at this infringement of their rites it brought one aspect of Christianity to a wider audience.
As the instability of the third century began to be checked by a succession of strong emperors, one in particular played the card that took Christianity from being a small religion amongst many to becoming the main player. As he was marching through the Italian countryside he saw what he thought were the letters CH and R (the Chi-Rho) and the words "in hoc signo vinces" (in this sign shall thou conquer) imposed on the sun. He took it to be a message from the Christian god. When he finally won his fight to control the once fractured Roman Empire, he believed he had done so with the help of the Christian God. On a more practical level he decided that to maintain and control the empire properly he would instigate just one state religion instead of having hundreds of self-governing temples and churches that were beyond his control. The religion he chose was Christianity.
From then on it became the dominant force and eventually the Popes became more powerful that the Emperors. By the time the empire crumbled, Christianity was established in all of its former provinces from Western Europe to Syria and North Africa to northern Britain. As the Dark Ages moved into the Middle Ages, Christianity became and has remained the most powerful force in the western world.