These were Jewish worship services, not Christian worship services. Paul went to the synagogues on the Sabbath because that is where he could speak to both large numbers of Jews and to the circumcised Gentile converts to Judaism.
Acts 13:43
Christians worshiped on the first day of the week, the Lord's day.
Acts 20:7,
1 Corinthians 16:2, Revelation 1:10
Numbers 29:35 On the eighth day you will hold a public assembly: you shall do no heavy work.
The eighth day (Sunday, the first day of the week) is the day the Christians have publicly worshiped God for 2000 years.
That said, the Christians did worship on other days of the week - the primary day of the week was always Sunday, the feast of the Resurrection, which we know because of the worship service in progress at the Cenacle at St. Mark’s house in Jerusalem at the third hour (9 AM) on a Sunday when the Holy Spirit descended, since Acts clearly says that there were 200 of the faithful present when that occurred.
However it is also the case that, ironically, the church that Sabbatarians appear to believe it is their religious duty to criticize, the Roman catholic Church, conducts more worship services on Saturday than any other denomination. All current and retired Roman Catholic priests are required to celebrate the Mass daily, and also the Liturgy of the Hours, and most churches have at least one public mass on Saturday, and recently Catholics have been permitted to attend a vigil mass on Saturday as a means of satisfying their Sunday worship obligation. The net result is that every Sunday, there are more masses celebrated in Catholic churches, cathedrals, priories and friaries, oratories, chapels, monasteries and convents than there are parishes of all Sabbatarian denominations combined - and all of these services, being masses, feature the Holy Eucharist, the Body and Blood of Christ our God given for the remission of sins and life everlasting (1 Corinthians 11:1-34), as a perpetual recapitulation of and participation in (anamnesis) of His voluntary sacrifice on the Cross, wherein God in the person of the Son and Word died in order to trample down death by death, and show us what it means to be human - death is swallowed up in victory as Christ arose from the grave on the First Day.
Thus, on Friday we commemorate that Christ remade humanity in His image, just as He made humanity in His image in Genesis 1 (it is not a coincidence that John chapter 1 parallels Genesis chapter 1, for Genesis 1 is a prophecy of the Passion of Christ our True God), on Saturday we commemorate Christ reposing in a tomb, and also by extension the mystery of Him who is uncircumscribable being contained in the womb of Our Glorious Lady Theotokos and Ever Virgin mary, and because of the Harrowing of Hell, a day to pray for our departed loved ones, and then on Sunday we celebrate Christ rising from the grave, and the light of the World to Come, the mystical Eighth Day of Creation.
The Sabbath is fundamentally prophetic, in that it was about the mystery of the Incarnation of Jesus Christ in the womb of the Theotokos and his repose in the tomb, and our own pending repose even as our soul shall either be with Christ in Heaven or not with Christ, depending on our probable eschatological outcome. Thus, traditional Christians properly observe this feast through the Vesperal Divine Liturgy on Holy Saturday in the Orthodox Church and equivalent services such as the Roman Catholic Paschal Vigil Mass (all of which originated in Jerusalem and are very similar, well, in the Roman Catholic Church it was nearly identical to the Orthodox liturgy until Pope Pius XII made some odd changes to the Paschal Triduum in 1955 - I don’t see what the point was of taking an ancient liturgy virtually identical to that celebrated by the Orthodox, which in the case of the Mass of the Presanctified on Good Friday contained elements in common with the Eastern Orthodox Presanctified Liturgy we attribute to Pope St. Gregory the Great, and replacing those liturgies with rescheduled liturgies very different from what was historically celebrated and what still is celebrated by the Orthodox.
Fortunately, Dr. Peter Kwasniewski has been encouraging Roman Catholics who are still able to celebrate the Traditional Latin Mass to use the pre-1955 Triduum, and thus the celebration of the ancient version of it is increasingly common. Alas, the SSPX, which is regarded as canonically irregular by many, and by some as schismatic, although I don’t think that’s the position of the Vatican, but I don’t know, only uses the final 1962 Tridentine missal, which of course has all the unpleasantness of the 1955 reforms.