The worship of Christians on Sunday is one of the most common Christian practices you will find as you travel the great diversity of Christendom. From the 100s to the early 1900s, it would be hard to find a Christian group for whom Sunday was not the focus of the Christian week. Although we can think of a few notable exceptions today, Sunday remains the "Lord's Day" for the vast majority of Christians.
The best known exception that comes to mind is of course the Seventh Day Adventists, considered by some to be a cult, mostly because they have so many beliefs that are idiosyncratic when compared to most other Christians. The one that gives them their name, and that would be most apparent to someone driving by one of their buildings, is that they do not worship on Sundays but on Saturdays. For example, you might see a sign advertising "Sabbath School" instead of "Sunday School."
But even though they stand so far outside the mainstream on some beliefs and practices, all of them connect in one way or another to the Bible. And it is precisely because the Seventh Day Adventists are trying to follow out the Protestant principle of "Scripture only" that they have felt free to jettison such well established Christian traditions as worshipping on Sunday and belief in conscious existence between death and resurrection. They have done so because they have focused on verses in the Bible other traditions tend to ignore or reinterpret.
For example, it is quite common among grass roots Christianity to forget that the Sabbath in the Old Testament was from sundown on Friday to sundown on Saturday. Every single reference to the Sabbath in the Old Testament refers to Saturday, including the fourth commandment to "Remember the Sabbath day, to make it holy" (Exod. 20:8*).  Many if not most American Christians equate this commandment directly with the practice of worshipping on Sunday, but they would be hard pressed to give you a biblical reason for the equation.
For example, every reference to the Sabbath in the New Testament also refers to Saturday. When Jesus heals a man on the Sabbath, he was healing a man on Saturday. When Colossians says not to let anyone pass judgment on you because of your observance of a Sabbath (Col. 2:16), it is referring to Saturday. Some have seen in the Greek wording of Matthew 28:1 a subtle indication that God shifted the Sabbath to Sunday at the point of the resurrection--"on the first of sabbaths." But all versions translate this idiom correctly as a reference to the first day of the week. Only by ripping the text from the way words were used at the time of Christ might one read it differently.
The New Testament thus does not equate the Old Testament Saturday Sabbath with Sunday. At least Paul's position on the issue of Jewish Sabbath observance is clear. Gentile believers are not obligated to keep the Jewish Sabbath. This is quite remarkable for some Christians to believe. Does not the Sabbath rule go back to creation, to the fact that God rested on the seventh day, Saturday (Gen. 2:2; Exod. 20:11)? Is not an Israelite stoned for picking up sticks on the Sabbath (Num. 15:32-36)? Did not Nehemiah stop the "evil" of buying and selling on the Sabbath around Jerusalem (Neh. 13:15-22)? Do we not find strong words in Isaiah, Jeremiah, and Ezekiel about keeping the Sabbath?
Indeed, all of these things are true. And it is just as true that Paul does not consider this teaching binding at least as it regards Gentile believers. "One person considers one day more important than another. Another considers all days the same. Let each be fully convinced in his [or her] own mind"* (Rom. 14:5). We have already mentioned Colossians 2:16: "Do not let anyone pass judgment on you because of food or drink or with regard to a festival, new moon, or sabbath."*
What Paul is arguing for here is generosity toward the practices of other believers. So "one person considers one day more important than another."* This is the Seventh Day Adventist and the person who believes Sunday is a Christian non-negotiable in one way or another. There is nothing in the New Testament that requires Christians to meet on Sunday. Paul would bid us not despise the Adventists for worshipping on Saturday, nor should we despise all the extra services various churches have devolved at other times and days of the week. If a church wants to have a second service on Saturday night, Paul says that is okay.
On the other hand, meeting on Sunday does seem to be an early practice, and where we might fault the Adventists is that they not only have decided to go with Old Testament practice. They have sometimes tried to argue that the rest of us should not meet on Sunday as we do. Some have tried to reinterpret what the "Lord's Day" is in Revelation 1:10, even though Christian writers like Ignatius clearly understand it to be Sunday, writing within fifteen years or so of Revelation.
Paul of course mentions setting aside contributions for the Jerusalem gift on the first day of the week (1 Cor. 16:2). Unfortunately, he does not tell us if the church met to worship on that day. Certainly in Acts Paul regularly visits synagogues until he is kicked out. But when it comes to the "Lord's Supper," he simply says "whenever you come together" (1 Cor. 11:17). We get the impression that they may have met more than once a week.
In any case, the practice of meeting together on Sunday is a well established Christian tradition that seems to have originated among the early Christians and clearly has continued throughout 2000 years of history. If God had a major problem with it, He has not bothered to tell us. It would seem to be a perfectly appropriate and acceptable way for Christians to remember Christ's resurrection. Sunday is the Lord's Day, the day most Christians have set aside to remember that Christ rose from the dead, to worship, pray, and often to eat the Lord's supper.
What is clearly not Christian is to despise other believers for their practice, unless they do not truly do what they do to honor the Lord (Rom. 14:6). The Adventists who worship on Saturday do as most Christian Jews of the first century surely must have, although many of them perhaps also met for at least a brief time on Sunday to remember the resurrection and look to Christ's return. The fundamentalists who have transferred the Sabbath laws to Sunday also do so as to the Lord. Many of them will not work on Sunday, nor will they buy and sell on the Lord's Day, which to them is the equivalent of the Sabbath. We should not despise them for these practices.
But they also must not despise those who have modeled themselves after perhaps most Gentile believers in the New Testament. Paul does not bind the Jewish Sabbath on them in any of its particulars. They meet together on the Lord's Day to remember the resurrection and look to Christ's return, but they work, buy, and sell on Sunday, as Paul probably did himself. If Paul refrained from these things, he would have done so on Saturday. They "consider all days the same" (Rom. 14:5), and they do it in honor to the Lord.
 This is the numbering of the commandments by Jews and the majority of Protestants. For Lutherans and Catholics, however, this would be the third commandment.