Christianity wasn't so much dealt with at the Imperial level until Valerian (reigned 253-60). While Christianity was certainly illegal throughout the whole Empire, Christians were typically dealt with at the local level, and so any persecutions were instigated by individual governors. Letter 10.97 from Pliny's collection of letters is representative. Trajan, the Emperor, writes to Pliny, who is serving as governor in Bithynia. Trajan writes:

1 Actum quem debuisti, mi Secunde, in excutiendis causis eorum, qui Christiani ad te delati fuerant, secutus es. Neque enim in universum aliquid, quod quasi certam formam habeat, constitui potest. 2 Conquirendi non sunt; si deferantur et arguantur, puniendi sunt, ita tamen ut, qui negaverit se Christianum esse idque re ipsa manifestum fecerit, id est supplicando dis nostris, quamvis suspectus in praeteritum, veniam ex paenitentia impetret. Sine auctore vero propositi libelli <in> nullo crimine locum habere debent. Nam et pessimi exempli nec nostri saeculi est.

Which is:

You have adopted the right course, my dearest Secundus, in investigating the charges against the Christians who were brought before you. It is not possible to lay down any general rule for all such cases. Do not go out of your way to look for them. If indeed they should be brought before you, and the crime is proved, they must be punished; with the restriction, however, that where the party denies he is a Christian, and shall make it evident that he is not, by invoking our gods, let him (notwithstanding any former suspicion) be pardoned upon his repentance. Anonymous informations ought not to be received in any sort of prosecution. It is introducing a very dangerous precedent, and is quite foreign to the spirit of our age.

It's important to note that Trajan does not endorse wild goose chases, or, worse, witch hunts. If people are Christians and they admit it, fine. But nobody needs to go out looking for trouble. He's probably reacting to the suspicions which characterized Domitian's reign (81-96). I think Tacitus' Annals has a similar sobriety when thinking about Christianity and he is a contemporary of Pliny and Trajan. Tacitus in 15.44.8 talks about the fact that the Christians were being destroyed by Nero not for the sake of public safety, but only because of Nero's cruelty.

Nevertheless, we see this sort of local persecution throughout the second century here and there. It happens in the late 170s in Lyons, France (Eusebius, HE 5.1), at which point Irenaeus of Lyons takes over as Bishop there. In 180, a group in N. Africa (in Scilli) were brought up on charges before the local provincial, Saturninus, who gave the Christians the option of recanting, but they persisted and were martyred.

While there was persecution under Septemius Severus (reigned 193-211), there was a period of de facto toleration of Christianity until the reign of Severus Alexander (reigned 222-235). This coincides with the fact that Christianity was making a lot of headway in more elite social circles. It could no longer be thought of as simply the religion of the peasants (it had always attracted some of the upper crust). Then comes Maximinus I (reigned 235-238), also known as Maximinus Thrax (he was Thracian), the first of the Barracks Emperors. This is a rather unstable time in the history of Rome, with Emperors being killed and replaced by men who had come up through the ranks of the military, who were then subsequently killed after being installed as Emperor. Maximinus was rather heavy-handed in his proscription of Christianity, but he was murdered 3 years into his reign, and the next decade or so saw Emperors who weren't in power long enough to worry about Christianity (there were 5 other Emperors just in 238 AD alone).

Moving ahead, when Decius defeats Philip I in 249, he implements some conservative reforms. Surely all this nonsense with so many bad Emperors is because the Empire had not been faithful to her identity. This meant cracking down on Christians again. The so-called "Decree of Decius" made every subject of the Empire responsible to sacrifice to the traditional gods and eat sacrificial meat. They were to obtain proof they had done so. This wasn't primarily an anti-Christian move, but it proved effective in exposing subversives. Since Christians could not participate in the Empire-wide sacrifice in January of 250, they were easily exposed and subsequently killed for being seditious. Decius is killed in 251, however, and Gallus rules for another 2 years, but then comes Valerian.

Eusebius tells us that Valerian was initially rather friendly towards the Christians (HE 7.10.3). But things took a turn for the worse and he issued edicts in 257 and 258 against Christianity. Prominent Christians (like Cyprian) were sought out and killed for refusing to worship the gods of the state, which, for the Romans, has not only religious implications, but civic ones as well. But Valerian is then captured by the Persians and his son, Gallienus, takes over for a few years. He repeals his father's edicts against the Christians and they're allowed to live in peace. Gallienus rules until 268. Things are pretty quiet up until Diocletian (reigns 284-305), who ushers in the greatest persecution of the Christians the Empire had ever seen. Diocletian not only expelled Christians from the army, put to death those who were too public about their Christianity, and imprisoned clergy, but also required Christians to turn over their Scriptures. Nevertheless, after Diocletian comes Maximian and then Constantius and the rest is pretty much history.