early Christians were first put to death by Jewish authorities, who used stoning, since other forms of execution were reserved for the (Roman) state authority. We have this from the accounts of the martyrdom of Stephen in Jerusalem, and Paul’s own account of the part he played in these activities.
The Roman’s began persecuting Christians in the reign of Nero when they became numerous enough to notice, when their public preaching and witness against the decadance and evil in the empire, the government, the society, and the emperor’s own household challenged the prevailing culture. The test for a person accused of promoting this sect was to acknowledge the emperor as a god which was fast becoming the state religion (even supplanting the importance of older Roman gods and godesses). This involved a public acknowledgement of the divinity of the emperor and eating food sacrificed to idols. Naturally the Christians refused and were persecuted. Nero used the growth of this sect to shift the blame for his own bad government onto the Christians when fire broke out and destoyed large parts of the city.
Persecutions under succeeding emperorers (who grew progressively more violent, more psychopathic and more short-lived) with a few notable exceptions of the few relatively benign reigns. When Constantine in the 4th century established Christianity as the official religion of the empire the persecutions ceased.
A.D .and I Claudius are two made for TV movies which depict fairly accurately the culture and climate of the world of the early Christians, and the cultural atmosphere in which the new religion grew in size, strength and influence, watered by the blood of the martyrs.