The first Christians were, like Jesus, Jews resident in Palestine who worshipped on occasion in the Temple in Jerusalem and weekly in local synagogues. Temple worship was a ritual involving sacrifice, occasionally including the sacrifice of animals in atonement for sin, offered to Yahweh. The New testament includes many references to Jesus visiting the Temple, the first time as an infant with his parents.
The early history of the synagogue is controverted, but it seems to be an institution developed for public Jewish worship during the Babylonian captivity when the Jews did not have access to the Jerusalem Temple for ritual sacrifice. Instead, to give a rough summary, they developed a daily and weekly service of readings from the Torah or the prophets followed by commentary. This could be carried out in a house if the attendance was small enough, and in many towns of the Diaspora that was the case. In others more elaborate architectural settings developed, sometimes by converting a house and sometimes by converting a previously public building. The minimum requirements seem to have been a meeting room with adequate seating, a case for the Torah scrolls, and a raised platform for the reader and preacher.
Jesus himself participated in this sort of service as a reader and commentator (see Gospel of Luke 4: 16-24) and his followers probably remained worshippers in synagogues in some cities. However, following the destruction of the Temple in Jerusalem in 70, the new Christian movement and Judaism increasingly parted ways. The Church became overwhelmingly Gentile sometime in the second century.
For the history of how services take place within a church, see worship or do a search on any particular religious denomination that you might be interested in.
The Syrian city of Dura-Europos on the West bank of the Euphrates was an outpost town between the Roman and Parthian empires. During a siege by Parthian troops in A.D. 257 the buildings in the outermost blocks of the city grid were partially destroyed and filled with rubble to reinforce the city wall. Thus were preserved and securely dated the earliest decorated church and a synagogue decorated with extensive wall paintings. Both had been converted from earlier private buildings.
The church at Dura Europos has a special room dedicated for baptisms with a large baptismal font.
A common architecture for churches is the shape of a cross (a long central rectangle, with side rectangles, and a rectangle in front for the altar space or sanctuary). These churches also often have a dome or other large vaulted space in the interior to represent or draw attention to the heavens. Other common shapes for churches include a circle, to represent eternity, or an octagon or similar star shape, to represent the church's bringing light to the world. Another common feature is the spire, a tall tower on the "west" end of the church or over the crossing.