The writings of the early Christians make it clear that Christians abandoned the Jewish custom of worship on the Sabbath (seventh day) and instead held the first day, which they called the Lord's Day, to be the sacred day of worship. One of the simplest explanations on the subject comes from Tertullian, around 200 AD:
But why is it, you ask, that we gather on the Lord's Day to celebrate our solemnities? Because that was the way the Apostles also did.
-- De Fuga in Persecutione, XIV: 11
This was not an innovation of the second century AD, as a full century earlier, (101 AD,) we find Ignatius saying:
Those who were brought up in the ancient order of things have come to the possession of a new hope, no longer observing the Sabbath, but living in the observance of the Lord's day on which our life was sprung by him and his death.
-- Epistle to the Magnesians, Chapter 9
And just in case there is any confusion as to the identity of the Lord's Day, we can turn to Justin Martyr. (140 AD)
And on the day which is called Sunday there is an assembly in the same place of all who live in cities or in country districts; and the records of the apostles, or the writings of the prophets, are read as long as we have time.
...
Sunday is the day on which we all hold our common assembly, because it is the day on which God, when he changed the darkness and matter, made the world; and Jesus Christ our Savior on the same day rose from the dead.
-- Apologies 1:67
And again from Justin Martyr, we see that Christians did not keep the Jewish holy days:
We neither accord with the Jews in their peculiarities in regard to food nor in their sacred days.
-- Apologies Sec. 21
So yes, it's clear from the historical record that the first-century Christians, such as Ignatius, had organized worship meetings on the Lord's Day (Sunday), and that they did not consider observance of the Jewish Sabbath important, it having been fulfilled by Christ with the rest of the Law of Moses.