The principle of the Sabbath rest started in Genesis, and is something that has been established since then.
Thus the heavens and the earth were finished, and all the host of them. And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made. And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.
Genesis 2:1-3
So the seventh day of the week was Saturday. We have kind of lost this concept in our culture, as the weekend feels like the end of the week, and therefore Sunday feels like the 7th day of the week. But technically Sunday is the first day of the week. Saturday is the 7th and is still the Jewish Sabbath up till this day. Interestingly, as an aside, the way that days started and ended in the Bible are different to our culture today too.
And God called the light Day, and the darkness he called Night. And the evening and the morning were the first day.
Genesis 1:5
The evening first, then the morning. Their days would start the evening before and end at the evening of the day. So in Jewish culture they would, and still do, end all their work by Friday evening, and observe the Sabbath from that evening, through the day on Saturday, then return to their usual work from Saturday evening. I know genuine believers that do the same. They will observe the Lord's Day from Saturday evening, and Sunday day, then return to normal work on Sunday evening. I also know some genuine believers who live in countries where the national Day of rest is another day, maybe a Friday, and this is the only day they are able to take off of work, so they observe the Lord's Day then.
So where did Sunday become the Christian day of worship from then? Well, first of all it was the day the Lord Jesus Christ rose from the dead.
In the end of the sabbath, as it began to dawn toward the first day of the week, came Mary Magdalene and the other Mary to see the sepulchre.
Matthew 28:1
All our hope is founded in Christ's death and the fact that He rose from the dead. We remember it each week and see how we should live our lives in the light of His death. Secondly believers were meeting together on Sunday and the Lord was appearing to them then. Exactly 1 week later, so the next Sunday by Jewish counting we read.
And after eight days again his disciples were within, and Thomas with them: then came Jesus, the doors being shut, and stood in the midst, and said, Peace be unto you.
John 20:26
I know it says 8 days later, but Jewish counting of days included the current day. So appears to be one more than we would count. Christ died on Friday afternoon, and rose on Sunday morning, so we in our culture would count it as He was dead for 2 days, but in Jewish culture it was counted as 3. So exactly a week later on Sunday, Christ appeared to them in the upper room. We also see other examples of believers gathering on Sunday in the new testament too.
And upon the first day of the week, when the disciples came together to break bread, Paul preached unto them, ready to depart on the morrow; and continued his speech until midnight.
Acts 20:7
and
Upon the first day of the week let every one of you lay by him in store, as God hath prospered him, that there be no gatherings when I come.
1 Corinthians 16:2
There aren't any examples in the new testament of believers now meeting for worship on Saturday.
Can true believers set aside another day of the week to worship God?
The Bible says:
Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:
Colossians 2:16
But this isn't for us to just choose based on preference. As I say, if you live in a country where another day is the only day of the week you can worship, then that is fine. And there can be true believers who do worship God on Saturdays. But check out the other things they believe, as often other things they believe don't match up with the word of God.
How you observe the Lord's Day matters as much as when you observe it. The Bible tells us to leave our work and our pleasures behind for one day.
If thou turn away thy foot from the sabbath, from doing thy pleasure on my holy day; and call the sabbath a delight, the holy of the Lord , honourable; and shalt honour him, not doing thine own ways, nor finding thine own pleasure, nor speaking thine own words: Then shalt thou delight thyself in the Lord ; and I will cause thee to ride upon the high places of the earth, and feed thee with the heritage of Jacob thy father: for the mouth of the Lord hath spoken it.
Isaiah 58:13-14
And
Remember the sabbath day, to keep it holy. Six days shalt thou labour, and do all thy work: But the seventh day is the sabbath of the Lord thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: For in six days the Lord made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the Lord blessed the sabbath day, and hallowed it.
Exodus 20:8-11
This is both paid work, and school work, and general things we could do on another day, like shopping. It's also doing things that are pleasures like going to football matches, or playing in one. Yes if there is an emergency reason, or something happens, then we are free to do things like go to the shop. One week, I forgot our Sunday lunch at home, and had to take the kids out to eat that Sunday. But it wasn't something I was aiming to do. We should be prepared and keep it as free as possible so we can keep it separate for the Lord. It's something that is there for our benefit.
And he said unto them, The sabbath was made for man, and not man for the sabbath:
Mark 2:27
We gain so much by regularly spending time focusing in Spiritual things, and what we gain spiritually especially forgiveness of sin helps all the rest of our lives.
One last thing. Don't just worship, read the Bible and pray on Sunday. Sunday is a day where we can especially focus on the Lord. But we need time with Him each day. So make sure you pray to Him each day, and think through and do His word. And worship Him for who He is and what He does each day.