The timeline is roughly (apologies if dates are inexact, but this is roughly correct from memory):
AD64: Nero perseuctes the Christians in Rome, who at that time were a tiny community
AD 65-249: Christianity is largely ignored and left to grow in peace. During this time, Rome enjoys a golden age under five virtuous emperors, then begins to disintegrate under the reign of various tyrants and opportunists.
AD 250-260: Rome is falling apart under various civil wars and there's two major state-led persecutions (by Decian and Valerian) of Christians. (The motivation being "rid the Empire of this impious cult which is weakening our social fabric").
AD 260-300: The "long peace of the Church", where Christianity is officially tolerated.
AD 300-310: Diocletian comes to power and starts a brutal series of persecutions to wipe out Christianity for good. It's from the Diocletian persecution that most of the images of brutal Roman suppression come from.
AD 310-325: Various civil wars, leading Constantine to come to power.
AD 325: Constantine establishes Christianity as the state religion.
So though there's one small persecution in 64, then two medium-sized persecutions in the 250s, then a massive persecution in the 300s, between these Christianity is either ignored or tolerated by the Roman state. There's evidence of mobs of Roman citizens attacking Christians independently of any state backing, but aside from that, Christians were largely left alone for a 200 year period.
This is the age of the 5 "enlightened" emperors - Trajan's attitude is informative. The Romans saw Christians as impious atheists (for not worshipping the Roman gods) and superstitious cultists (for following a novel religion), and they were troubled by the growth of this sect; Trajan took it for granted that Christians who refused to worship the Roman gods should be punished. But, he also states that actively seeking out Christians would be "out of keeping with the spirit of our age"- a weird Roman kind of tolerance, but there you go.
"You observed proper procedure, my dear Pliny, in sifting the cases of those who had been denounced to you as Christians. For it is not possible to lay down any general rule to serve as a kind of fixed standard. They are not to be sought out; if they are denounced and proved guilty, they are to be punished, with this reservation, that whoever denies that he is a Christian and really proves it--that is, by worshiping our gods--even though he was under suspicion in the past, shall obtain pardon through repentance. But anonymously posted accusations ought to have no place in any prosecution. For this is both a dangerous kind of precedent and out of keeping with the spirit of our age."
(Trajan's letter to Pliny here: http://faculty.georgetown.edu/jod/texts/pliny.html)