From the point of view of the Romans Christianity started out as a cult branch of Judaism. Christianity spread the same way all early religions did. They used Roman roads to spread the message throughout the Roman empire and Middle East. Of course the first and only converts for the first couple hundred years were poor people because the idea of rewards in the afterlife sounded awesome to poor people.
At first Rome turned the other eye but after Christians started doing things that seemed unpatriotic to the Romans, such as refusing to pay tribute to the emperor or praying to the Roman gods, Roman authorities began to get frustrated with Christians. Although the exact numbers are largely exaggerated today, a few Christians were thrown in Colosseums for sport but not as many as you'd think. Christians were blamed for a lot of Rome's misfortune such as the time Rome's capital caught on fire. Everybody blamed Christians for the fire and many people of the time believed that the Christians purposely started the fire to kill nonbelievers but many modern Historians think that Nero might have actually been the one to start it. Christians still spread their religion to the poorer citizens in Rome's backallys and ghettos.
After a while though as more and more citizens became Christians Rome tried to become politically correct and became overly complacent to Christian's demands. They allowed Christians to refuse military service. After the emperor himself became Christian he made a Roman version of Christianity (in other words "sugarcoated version") the state religion. He allowed Christians to refuse worship of foreign gods. Now it wasn't just the poor who were targeted for conversion. Many pagans and non-Christians suddenly experienced hardship as a Romanized version of the faith took hold. Polytheists were forced to convert or killed. Pagans were discriminated against. Now the peaceful religion that Rome once saw as a backwater version of Judaism was the strongest faith in the west.
However, their non-Christian neighbors had it pretty rough as the religious powers of Rome began cutting off ties with non-Christian peoples. To answer you're question peacefully, Christianity gained followers at first by targeting the poorer citizens of society. Later on they gained followers by the sword of Rome. There are exceptions to the rule. St. Patrick gained followers peacefully in Ireland and Egypt gained followers peacefully enough during Rome's reign. But for the most part, after the rise and fall of Christian Rome, the mindset was "convert or shut-up". I just realized how big this answer is. You probably won't read this /: