I touched upon Christianity in the Roman Empire in a question about Marcus Aurelius' policies against Christians.

To answer to your question, it really depended where you were. The Romans were tolerant of other faiths but that didn't mean they respected them. They considered the practices of other religions as bizarre and preposterous. For example, Plutarch thought the Sabbath was ridiculous since Jews refused to do anything on the Sabbath, even fighting in self defense. He regarded it as a religious holiday ordained to enforce laziness.

Christianity was seen the same way and for a long time it regarded as one of the many Jewish sects that existed during this time. Since most of the first converts were Jews, the earliest Christian persecution was Jews suppressing other Jews. It wasn't seen as a Roman problem until Gentiles began converting in large numbers.

Christianity declared that there was no inherent difference between men, women, cultures, masters, or slaves. It was founded by a desert preacher who used to be a carpenter and was sentenced to crucifixion, a very public and humiliating form of capital punishment that was reserved only for the worst of criminals. It was monotheist and exclusivist. All of this put it at odds with Roman culture.

But the level of persecution varied depending on the provincial governor and the temperament of the local people. Though Christians held some very radical beliefs, their lives continued as normal. The apostles encouraged people to continue being part of the secular community. Lucius the baker didn't suddenly stop making bread after becoming a believer. Julia the merchant's wife didn't stop going to the market. Life continued as normal and Roman governors rarely took action unless a mob demanded it. If you lived in one of these more lenient communities, it was unlikely you'd face any serious legal action.

But if you lived in a place like Lyons in the first century, you would've probably wound up dead. The population of Lyons took it upon themselves to start banning Christians from public areas like the baths and the forum. After a significant amount of pressure, the governor arrested the Christians and dealt out various punishments. Some were condemned to be eaten alive by animals, some were tortured, and some were imprisoned until their deaths. Even those who recanted their faith during their trials were not spared. In some Roman towns, apostasy was all that was required to release someone accused of being a Christian, and sometimes they were released even if they refused to disavow their beliefs.

Persecution on the imperial level began to ramp up during Marcus Aurelius' reign. When Decius took the throne, all bets were off. That was when Christians throughout the empire were brutally persecuted under imperial authority. Most of the decision making still lied in the hands of local officials but the new edicts made it easier to prosecute Christians and the rapid growth of the religion among non-Jews scared the average Roman.

As I mentioned earlier, Christianity didn't elicit much sympathy from conservative Romans. Christians worshiped a convicted criminal, refused to participate in public sacrifices, rejected the imperial cult, held their services in private, and practiced some customs that the Romans found really creepy. The sacrament of Holy Communion led to rumors that Christians were cannibals. How different Roman communities dealt with that perception was a case by case basis in 75 CE.