Obviously, at the time of Jesus' Crucifixion, newly converted Christians were being actively hunted and persecuted. But in the ~100 years, how were they able to pick up the pieces and how did Christianity take root and spread to Rome, to the point that Catholics are known as Roman Catholics?
Note: I'm not asking about Church teachings or anything. I was raised Catholic and have never really learned about the gap between Jesus' death and current times.
Perhaps you should start reading the so-called "Church Fathers"' writings. I'm talking about Polycarp, Iraeneus, Justin Martyr, Jerome, Augustine... these were bishops that lived between the 2nd and 8th centuries.
I saw these collections in the University Library. They were called "Antenicene Fathers" (who lived before the council of Nicea, "Nicene Fathers" (who lived around that council), and "Postnicene Fathers".
But take it with a grain of salt. Remember that those are official Church leaders speaking for the community. Perhaps you should also consider studying other branches of Christianity that existed along those times, like the gnostics. In that line of thought, I should also recommend you reading Bart D. Ehrman's "Lost Christianities" (also on my to-read list), which explains what different branches of Christianity were there, and how Catholics became the official Church.