Forgive my simplistic understanding, but over the last 40 years of my Christian life I have always been taught that the "church" started in Acts 2 when the Holy Spirit indwelt believers. The rest of Acts, as I understand it, describes the "Word of God" (the preaching that Jesus of Nazareth is God's one and only human born Son, that he died on the cross in our place, and that three days later God raised Him from the dead, and that He alone is the only way to have eternal life with God) moving from Jerusalem into the rest of the world.
It is also my understanding that Acts, along with many of the Epistles, give understanding into the nature of local churches at the time.
So, it appears to me that the "Word of God", not the Bible, came and made His dwelling among us, and that through His death burial and resurrection we can have eternal life with God. Where a group of like minded believers gather together for worship, fellowship, encouragement, and support, etc., there the church is. This started a few weeks after Jesus' death burial and resurrection, and continues to this day.
A few years after the beginning of the church of Christ, the four Gospels were written, along with the rest of the books of what we now consider the New Testament. These were copied and passed around the various local churches in the near east, and finally compiled by the third century. This is what we now consider the "New Testament" of the Bible.
As I understand it, Paul and Peter traveled to Rome and established local churches in that area, and one of the local churches Peter helped start grew and, after a few hundred years, turned into what we now recognize as the "Roman Catholic" church.
It is also my understanding that "catholic" means universal. It comes from the Greek word katholikos - which means "universal".
This means there is a distinction between "catholic" which means "universal" and "Roman Catholic" which is an organization that specifically claims its origins to be in Rome of the first few centuries AD.
So, in answer to the OP's original question, the "church" (as defined in Acts) came first. (But this is in no way the "Roman Catholic" church.) Over the next generation, the writings which we now consider to be parts of the New Testament were written. Over the next two to three centuries, the writings were compiled and defined as "The New Testament". (How they were compiled or under what authority they were selected is beyond the nature of this post.)
Also, over the course of the first few centuries AD, the local church in Rome grew and grew and eventually proclaimed itself to be the mother church of all churches. I am not saying whether they were justified in doing this or not. But, that is still their claim.
At least, this is my current understanding. I am open to correction and clarification as appropriate.