I've answered the question about early Christian conversation rates before, so I put the link to it here.

However, here is a clip from that answer:

Rodney Stark, a sociologist, wrote a book called The Rise of Christianity: How the Obscure, Marginal Jesus Movement Became the Dominant Religious Force in the Western World in a Few Centuries and discuss this very topic. Now I want to explain that Stark is a sociologist, not a historian and this book was actually attacked for several reasons by many scholars. However, he did make one point that was actually agreed upon by other historians. The first is that Christianity, started out quite small, possible as little as 20-30 people by the time that Jesus was executed. However, it can be reasonably assumed that a steady growth rate of about a 40% increase to these followers happened (on average) every decade until the fourth century, where it exploded even faster.

By the time Christianity was endorsed by Constantine around 313 CE, roughly 3 million people (5% of the Roman population) were Christian.

That said, Christian tradition teaches that Christian's came under heavy persecution throughout their early existence. Being martyred and being thrown into gladiatorial battles is often envisioned and cited whenever I ask my students (I'm a teaching assistant at a university) what they thought the early years were like. However, this isn't the most reasonable assessment nor is this belief held by most scholars.

A truly excellent book on this topic was by historian Candida Moss, whom wrote The Myth of Persecution: How Early Christians Invented a Story of Martyrdom. Here, Moss (a professor of New Testament Studies at Notre Dam) argues that there was never a wide-spread persecution of Christians as most people believe. Of the over 250 years of Early Christian history, Moss presents evidence that at most there were only 12 years (at most) where people were harshly persecuted for their faith by orders of the Roman Empire (none of them coming in the first century, during the early conversion times). For it's first few decades, Christians probably went relatively unnoticed. Small churches were known to have sprung up throughout the eastern Mediterranean and even at least one in Rome by 70 CE (this is about when and where Mark wrote his gospel), but it is unlikely Christians faced any real threat yet since they weren't converting huge percentages of people in their populations.

As Christianity became more popular, most likely by the end of the first century, it is likely that this is when they started facing harsher criticism. Most were simply scorned or disliked, similar to hows Jews were treated once they left the Levant. For most early Christians, "the vast majority of Christians never stood before a Roman judge, paid a fine, or experienced torture," [Moss, page 129] however, some did. Moss (and other historians) have pointed out that persecution did take place in several locations around the Roman Empire, but again, this is more the exception to the rule. The reason for this is simple, "For almost all of the first century, it’s unclear that Roman emperors even knew that Christians existed," [Moss, page 139] meaning that you can't persecute and hate something that you didn't even know existed.

I will add that there were some specific areas and period of persecution where select numbers of Christians were tortured or even killed, but this didn't happen until the second century, and was rare and didn't last for very long. For the most part, Christianity was a small sect that didn't expand considerably until Constantine, and for the most part, they didn't really pose as a serious threat to Rome.

edits: spelling