Why do we worship on Sunday?

In the Old Testament, God stated, "Remember the Sabbath day, to keep it holy. Six days you shall labor and do all your work, but the seventh day is a Sabbath of the Lord your God; in it you shall not do any work, you or your son or your daughter, your male or your female servant or your cattle or your sojourner who stays with you," (Exodus 20:8-10, NASB). It was the custom of the Jews to come together on the Sabbath, which is Saturday, cease work, and worship God. Jesus went to the synagogue on Saturday to teach (Matt. 12:9, John 18:20) as did the apostle Paul (Acts 17:2; 18:4; ). So, if in the Old Testament we are commanded to keep the Sabbath and in the New Testament we see Jews, Jesus, and the apostles doing the same thing, then why do we worship on Sunday?
First of all, of the 10 commandments listed in Exodus 20:1-17, only 9 of them were reinstituted by in the New Testament: (six in Matt. 19:18, murder, adultery, stealing, false witness, honor parents, and worshiping God; Rom. 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 said that He is the Lord of the Sabbath (Matt. 12:8).
In creation God rested on the seventh day. But, since God is all powerful, He doesn’t get tired. He doesn’t need to take a break and rest. So, why did 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, of focus on God. In this, 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 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 Jesus’ atonement, we no longer are required to keep the Law. We are not under Law, but grace (Rom. 6:14-15). The Sabbath is fulfilled in Jesus. He is our rest. We are not under obligation, by Law, to keep it and this goes for the Sabbath as well. It is not a requirement that we keep the Sabbath. If it were, then we would still be under the Law. But, we are not.

Evidence of the Change of Days can be Seen in the NT

Within the New Testament is ample evidence that the seventh day Sabbath is no longer a requirement.

Rom. 14:5-6, "One man regards one day above another, another regards every day alike. Let each man be fully convinced in his own mind. 6 He who observes the day, observes it for the Lord, and he who eats, does so for the Lord, for he gives thanks to God; and he who eats not, for the Lord he does not eat, and gives thanks to God."
The entire section of Rom. 14:1-12 is worth careful study. Nevertheless, the instructions here are that individuals must be convinced in their own minds about which day they observe for the Lord. If the seventh day Sabbath were a requirement, then the choice would not be mans’, but God’s.

Col. 2:16-17, "Therefore let no one act as your judge in regard to food or drink or in respect to a festival or a new moon or a Sabbath day— 17 things which are a mere shadow of what is to come; but the substance belongs to Christ."
Notice here that time sequence mentioned. A festival is yearly. A new moon is monthly. A Sabbath is weekly. No one is to judge in regard to this. The Sabbath is defined as a shadow, the reality is Jesus. Jesus is our Sabbath.

Acts 20:7, "And on the first day of the week, when we were gathered together to break bread, Paul began talking to them, intending to depart the next day, and he prolonged his message until midnight."
The first day of the week is Sunday and this is the day the people gathered. This passage can easily be seen as the church meeting on Sunday. It has two important church functions within it: breaking bread (communion) and a message (preaching). Additionally, Luke did not use the Jewish system of counting days: sundown to sundown. He used the Roman system: midnight to midnight. This is a subtle point that shows the Jewish Sabbath system was not the one utilized by Luke.

1 Cor. 16:1-2, "Now concerning the collection for the saints, as I directed the churches of Galatia, so do you also. 2 On the first day of every week let each one of you put aside and save, as he may prosper, that no collections be made when I come."
Notice here that Paul is directing the churches to meet on the first day of each week and put money aside. It would seem that this is tithing. So, the instructed time for the church to meet is Sunday. Is this an official worship day set up by the church? You decide.

Rev. 1:10-11, "I was in the Spirit on the Lord’s day, and I heard behind me a loud voice like the sound of a trumpet, 11saying, "Write in a book what you see, and send it to the seven churches: to Ephesus and to Smyrna and to Pergamum and to Thyatira and to Sardis and to Philadelphia and to Laodicea."
The New Bible Dictionary says regarding the term, ‘The Lord’s Day’ in Revelation 1:10: "This is the first extant occurrence in Christian literature of heµ kyriakeµ heµmera. The adjectival construction suggests that it was a formal designation of the church’s worship day. As such it certainly appears early in the 2nd century (Ignatius, Epistle to the Magnesians, 1. 67).
In many churches today, the term "The Lord’s Day" is used to designate Sunday, the same as it was in the second century.

I hope this is evidence enough to show you that the Bible does not require that we worship on Saturday. If anything, we have the freedom (Rom. 14:1-12) to worship on the day that we believe we should. And, we no one should judge us in regard to the day we keep. We are free in Christ, not under law (Rom. 6:14).