Why do Christians worship on Sunday, while the Old Testament law (as given in the 10 Commandments) set aside Saturday as the day of worship?
It was the custom of the Jews to come together on the Sabbath, which is Saturday, cease work, and worship God. As we study the New Testament, we observe that of the 10 Commandments listed in Exodus 20:1-17, only nine of them were reinstituted in the New Testament. (Six in Matthew 19:18, murder, adultery, stealing, false witness, honor parents, and worshiping God; Romans 13:9, coveting. Worshiping God properly covers the first three commandments). The one that was not reaffirmed was the one about the Sabbath. Instead, Jesus told us that He is the Lord of the Sabbath (Matt. 12:8).
The basis of the origin of the Old Testament command comes from creation when God rested on the seventh day. But, since God is omnipotent, He doesn’t get tired. He doesn’t need to take a break and rest. So, that causes us to stop and ask, why does it say that He rested? The reason is simple: Mark 2:27 says, "The Sabbath was made for man, and not man for the Sabbath." In other words, God established the Sabbath as a rest for His people, not because He needed a break but because we are mortal and need a time of rest to put our focus on Him. By taking this period of rest, our spirits and bodies are both renewed.
The Old Testament system of Law required keeping the Sabbath as part of the overall moral, legal, and sacrificial system by which the Jewish people satisfied God’s requirements for behavior, government, and forgiveness of sins. The Sabbath was part of the Law in that particular sense. In order to "remain" in favor with God, you had to also keep the Sabbath. If it was not kept, then the person was in sin and would often be punished (Ezekiel 18:4; Rom. 6:23; Deut. 13:1-9; Num. 35:31; Lev. 20:2, etc.).
But with the fulfillment of Jesus’ atonement, and our justification by faith (Rom. 5:1), we no longer are required to keep the Law and hence the Sabbath which was only a shadow of things to come (Col. 2:16-17) for our salvation. We are not under Law but grace (Rom. 6:14-15). The Sabbath is fulfilled in Jesus because in Him we have rest (Matt. 11:28).
While the moral principles still are binding, we are not under the command to rest and worship on Saturday by many observations for the New Testament Church. Jesus resurrection was on the first day of the week (Matt. 28:1). He continued to appear on succeeding Sundays (John 20:6), and the descent of the Holy Spirit was on Sunday (Acts 2:1). The pattern of the early church was to gather to worship on Sunday (Acts 20:7; 1 Cor. 16:2).
Prayer: Father, I not become entangled in confusion over days and restrictions that are not binding upon me. You want my love, and I will seek to love You with my whole being. Amen.
 Slick, M. (n.d.). Should we keep the Sabbath or not? Retrieved February 28, 2017, from CARM: https://carm.org/should-we-keep-sabbath-or-not
 See more at Geisler, N. L., & Howe, T. (1992). The Big Book of Bible Difficulties. Grand Rapids, MI, USA: Baker Books. Pages 77-78, 329-330.