The Christians during the apostolic era (basically the first century, from about 35 to 100 A.D.) kept Sabbath on the seventh day of the week, more or less what we now call Saturday. (More or less" because the Bible Sabbath begins and ends at sundown whereas Saturday begins and ends at midnight). For the first 300 years of Christian history, the Christian religion was an illegal religion, but Judaism was a legal religion. During the apostolic era, Christians found it convenient to let the Roman authorities think of them as Jews. Sabbath observance is quite visible. Thus, during those early years of Christianity observance of Sabbath helped the Christians to be identified with Judaism, which gained them legitimacy with the Roman government. However, the Jews rebelled against Rome, and the Romans put down their rebellion by destroying Jerusalem in 70 A.D. and again in 135 A.D. Obviously, the Roman government's suppression of the Jews made it increasingly uncomfortable for Christians to be thought of as Jews. So instead of trying to be thought of as Jews, some Christians in the early second century tried to distance themselves from Judaism, and observing a different day was one easy way to do that. And that was a very convenient way to do it. Many pagans observed the first day of the week in honor of their sun god. By switching to observing the Sabbath on the first day of the week the Christians accomplished two things: They distanced themselves from Judaism, and they made it easier for pagans to become Christians. Of course, their observance of the Sabbath on Sunday was without biblical support. They just did it. This is not to say that all Christians suddenly started keeping the first day of the week. The movement toward keeping Sunday began quite small.
The earliest positive evidence for a Christian worship service on Sunday is sometime between about 115 and 135 A.D., probably in Rome. We also know that the church in Rome was particularly active in promoting Sunday observance. For several centuries, some Christians kept Sabbath on Saturday and some kept it on Sunday. By the 500s, Sunday observance was pretty universal. If you are interested in learning more about this topic, the book "From Sabbath to Sunday" by Samuele Bacchiocchi has a very detailed history. You can purchase this book online by