Answer:
Christianity spread throughout the Roman Empire in a series of steps.
We can start with the Day of Pentecost, the very first day of the Church, when Peter declared to the thousands gathered in Jerusalem for that festival that salvation, leading to eternal life, was available to all who repented of their sins and were baptised in the name of Jesus Christ. He told them that this promise was available to everyone, whether they were present that day, or in that place, or were far off in time or place. Three thousand became Christians on that day. They were drawn from a crowd of people who came from all over the Roman Empire, and from the Parthian (Persian) Empire as well.
A few years later, most of the early Christians fled from Jerusalem, since the authorities there were offended by the preaching of one of the Christian leaders, Stephen. One of the destinations of the fleeing Christians was the Syrian city of Antioch. It was in Antioch that Christianity began to be actively preached to non-Jews. This eventually led to the first organized missionary journey, conducted by Barnabas and Paul, and saw the Christian message go to Cyprus and then to southern Galatia, in Anatolia (modern Turkey). New churches were established in both regions. Further missionary journeys, by Paul and Silas, saw Christianity being spread in Greece and in the Roman province of Asia (also in Anatolia). Other missionaries took the Christian message to other places, such as Bithynia, Pontus and Cappadocia (also in Anatolia), and individual Christians certainly took the Christian message to Rome.
So, within a generation of the Church starting, Christianity had already gained a foothold in many provinces of the Roman Empire, and certainly in all the provinces of the eastern half of the Roman Empire.
The next two hundred years saw the Christian Church expand and grow from these early roots. We have very little information about how this happened, but we have a few precious fragments, such as the account of the persecution of Christians in two cities on the Rhone river (in modern France). This is discussed in the related question, shown below, "What caused the spread of Christianity to go north of the alps?"
In Bithynia (in north-western Anatolia), at the beginning of the second century, the Roman governor, Pliny, discovered that the old cults were being deserted, and Christianity was widespread amongst the ordinary people. He claimed he had been able to suppress the Christian cult, and to encourage the people to return to the old cults, through a widespread policy of torture and execution of those most intractable followers of the Christian religion. Emperor Trajan supported his actions, but warned against a policy of seeking out Christians too vigorously.
Meanwhile, supporters of the traditional religions, and particularly the philosophers, were becoming aware of the challenge posed by Christianity. They met it in various ways. The best known of these responses is the work, The True Word, written by a Platonic philosopher in the middle of the second century, and embedded in a rebuttal written by the Christian philosopher, Origen, in the middle of the third century.
Origen's influence is rather difficult to assess, however, it appears that most of the bishops of the provincial capitals in Anatolia in the second half of the third century were influenced by his writing and teaching.
We know less than we would like to know about Christianity in the second half of the third century. Yet we can discern that the influence of Origen's teaching probably saw Christianity gaining intellectual ascendancy in Greece, Anatolia and Syria / Palestine. It was also gaining a measure of popular support in those places as well as in Egypt.
This period was bracketed by the two great periods of persecution in the history of the Church. The first was launched by Emperor Decius in 250, and tried again by Emperor Valerian in 257. The second was launched by Emperor Diocletian in 303. It is difficult to avoid the conclusion that the Church in the eastern half of the Roman Empire grew enormously between these two horrendous events.
In the western half of the Roman Empire, the Church was also growing, but it does not appear to have reached anywhere near the strength and importance of the Church in the eastern half of the Roman Empire. However, Constantine's conversion to Christianity, which can be reasonably dated to 312, changed everything, with the Christianity's status in the western half of the Empire being changed from a mostly tolerated religion, and sometime persecuted religion, to a favoured religion.
In 313, the emperor in the eastern half of the Roman Empire, Licinius, joined with Constantine in publishing the Edict of Milan, which granted Christianity formal tolerance throughout the Roman Empire. These colleagues in the Empire were also rivals, and in 324 Constantine defeated Licinius, and began to rule the entire Roman Empire on his own. Christianity was then favoured throughout the whole of the Roman Empire, in both the west and the east.
Christianity was already on the move before Constantine took power; now with state support it grew from strength to strength. The great Christian historian, Eusebius, admitted that he believed that many conversions to Christianity were for the wrong reasons. On the other hand, we have sufficient evidence to show that the free spread of Christianity resulted in many genuine conversions, and more significantly for this topic, the gradual phasing out of the old cults as a central part of the people's lives, with the notable exception of Rome itself.
There is no doubt that a large measure of self-interest was involved in some conversions to Christianity. One can understand that this applied particularly to the curial class. These were the rich people who were required to fund basically all the civil functions of the state, and kept the cities operating. Initially, it was a simple change, since those members of the curial class who became Christian priests were no longer required to meet these obligations. This followed the previous practice of extending exemptions to those members of the curial class who took on other civic obligations, such as being priests of the traditional cults, taking responsibilities in the military, or becoming a senator. However, it was not really equivalent, and as a result, in 329, Constantine required bishops and clergy to be generous towards the poor. He also restricted the entry of the curial class into the priesthood.
Missionary activities within the Roman Empire continued during the fourth century, with most significant being the mission of Ulfilas to the Goths on the borders of the Roman Empire, which eventually led to large numbers of Goths and other Germanic peoples living within the Roman Empire becoming Christians. As it turned out, these Germanic Christians were largely estranged from their fellow Roman and Greek Christians because the creed that Ulfilas taught them was not the Nicene Creed, but rather a creed that distinguished between the persons of the Trinity.