Praising God, and having favour with all the people. And the Lord added to the church daily such as should be saved.
-Acts 2:47
Answer: When we say a "church", it doesn't literally mean a building nor a place of worship but a congregation of the saints. The founding of the very first Christian church was mentioned in the Bible. It can be read on the first and second chapters of the book of Acts of the Apostles. It was the church in whom the Twelve apostles themselves have founded and established.
The day prior to the ascension of Jesus, he gave a commandment to the apostles to stay in Jerusalem. They'll have to wait until the day when he will send th Holy Spirit as he was promised earlier (Luke 24:49; Acts 1:4-5). With one accord, the apostles, Mary, the brothers of Jesus and with other disciples assembled themselves together in an upper room of a certain house in Mount Olives. Together, they were praying and later decided to choose someone who will replace the apostleship forsaken by Judas Iscariot. The lots fell on Matthias and was numbered among the Twelve Apostles (Acts 1:24-26).
The First Church in Jerusalem
In the day of Pentecost the Holy Spirit was poured down from the heaven and filled the house wherein the disciples of Christ were gathered together. There appeared unto the people something like a tongue of fire. Then suddenly, everybody present in the room were speaking different languages (Acts 2:1-4). The multitude that witnessed this event were amazed and confounded. Later, with Peter's exhortation, many people believed and were baptized. On that single day alone, the small number of disciples were added with 3000 more souls (Acts 2:41). Believers were gathered together in one accord, praying, giving heed to the doctrines of the apostles and their miraculous woks and even sold their own possessions to be shared by all (Acts 2:42-46). Those early days of Christianity was revered as one of the most inspiring event written in the Bible. Finally, God added more converts to the church daily, those who were saved (Acts 2:47).
Aside from this story in the Bible, there is another claim that is still debatable today. The Catholic church proclaimed that the first church was actually established in Rome and not in Jerusalem. According to Catholic traditions, prior to the rise of the churches in Jerusalem and Antioch, Peter and the other apostles were already traveling and preaching the gospel to the Jewish communities outside Palestine. One tradition recounts the missionary works of the apostle James (the brother of John) in Spain. Peter, on the other hand, has made an earlier missionary works which led him in founding and establishing the very first church in Rome. This traditional claims, however, was never been widely accepted both by historians, scholars and theologians. Since it doesn't only negates totally the story written in the book of Acts but it also has no known written or historical evidence.
