Short answers, pulled from the story.
Christianity began as a Second Temple Jewish movement in the 1st century AD in the Roman province of Judea. According to the Acts of the Apostles, the Jerusalem church started at Pentecost with around 120 believers. Within ten years of Jesus's death, apostles had established communities from Jerusalem to Antioch, Ephesus, Corinth, Alexandria, and Rome.
Paul was personally responsible for bringing Christianity to Ephesus, Corinth, Philippi, and Thessalonica. He relaxed traditional Jewish requirements for Gentile converts, which opened the Christian movement to a far larger population. According to Krister Stendahl, Paul's central concern was the inclusion of Gentile Torah-observers into God's covenant, not the individual conscience of sinners.
Bart D. Ehrman identifies five factors: the promise of salvation and eternal life for everyone, stories of miracles depicting the Christian God as powerful, hope for the lower classes, the requirement to abandon all other gods (which built strong communal identity), and the Roman household structure that meant converting the head of a household often converted his entire family and slaves. By 300, Christians made up roughly 10% of the Roman population; by 350, that figure had risen to 56.5%.
There was no empire-wide persecution of Christians until the reign of Decius in the 3rd century. The Edict of Serdica, issued by the emperor Galerius in 311, ended the Diocletianic persecution in the East. The Edict of Milan, issued by Constantine the Great and Licinius in 313, legalized the Christian religion throughout the empire.
Armenia became the first state to declare Christianity its official religion in 301 or 314, while the faith was still illegal inside the Roman Empire. The Armenian Apostolic Church traces its founding partly to Gregory the Illuminator of the late 3rd and early 4th centuries.
The persecution, which tradition dates from 339 to 379, was primarily political rather than religious. Around 315, a letter from the emperor Constantine to the Persian shah Shapur II, praising Persian Christians as people worthy of imperial protection, made Shapur suspect Christians formed a fifth column loyal to Rome. Shah Shapur II ordered double taxation on Christians and eventually the execution of clergy, including Bishop Simon, who was put to death in Susa in the year 344.