The New Testament makes it clear that the observance of a particular day was not imposed as a binding obligation.
Romans 14
5 One person regards one day above another, another regards every day alike. Each person must 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.
This passage makes it clear that there was some freedom in the matter of special days.
Colossians 2
16 Therefore no one is to 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.
Here, Paul commanded the church not to allow anyone to act as their judge in regard to sabbath days.
Galatians 4
9 But now that you have come to know God, or rather to be known by God, how is it that you turn back again to the weak and worthless elemental things, to which you desire to be enslaved all over again ?
10 You observe days and months and seasons and years.
Again, Paul warns against going back under the Law by insisting on the legal requirement of special days.
The records that remain in the New Testament show that the first day of the week soon became a day of worship. When Paul wanted to collect an offering from the church at Corinth, he asked them to gather the money on the "first day of the week".
1 Corinthians
2 On the first day of every week each one of you is to put aside and save, as he may prosper, so that no collections be made when I come.
And when he wanted to meet with the believers at Troas, the gathering took place "on the first day of the week, when we were gathered together to break bread".
Acts 20
7 On the first day of the week, when we were gathered together to break bread, Paul began talking to them, intending to leave the next day, and he prolonged his message until midnight.
There is no Scripture passage that specifically teaches that the sabbath has been transferred from one day to another. It seems most likely that the shift from Saturday to Sunday was gradual, and took place along with the change from a mostly Jewish church to a mostly Gentile one. The early church fathers generally viewed sabbath as a Jewish observance, and the Lord's Day, Sunday as identified by John in The Revelation, as the proper Christian observance.