Early Christians met at different times but their principal day of meeting was the first day, Sunday. Important activities went on during that day in the Church (Acts 20:7; 1 Corinthians 16:2.) Contrary to a lot of popular thinking though, Sunday did not become the new Sabbath.
We know in the New Testament many Jewish Christian’s including Paul continued to observe the Sabbath attending the Synagogue at that time. If the day of the Sabbath had shifted why did they continue to observe it on the old day. Furthermore Paul condemned others for criticising people on the observance of the Sabbath (Colossians 2:16.)
The first day represented something new. This was quite appropriate being a reference to Genesis and the first day since they were preaching a new creation. All the Evangelists testify that the Lord rose from the dead on the first day of the week. The resurrection is the crux of the Christian faith and so worship on the first day, the Lord’s Day, was of great symbolic value. Sunday had the added advantage that all believers, both Jew and Gentile, could participate. Sabbath regulations could only be properly observed if you belonged to the Jewish community. However, the Church is about breaking down such distinctions so we are one new man in Christ (Galatians 3:28.)
Perhaps the most disturbing thing about the idea of the Sunday not being a new Sabbath is what it means for the Ten Commandments. We are usual taught that the Decalogue is moral law so it still applies to the New Testament believer. The idea of law in the Bible, as I have explained else where, is not nearly as simple. When we read Hebrews 3 and 4 we see the meaning of the Sabbath is spiritualised and fulfilled in the Messiah. This is consistent with the new Exodus language and promises of rest found in the rest of New Testament.
Besides I believe any modern attempt by Christians, including the Adventists, to implement the Sabbath practically ends up as a distortion or a parody of what real Sabbath and Torah observance looks like.
We simply have no evidence suggesting early Christians thought of Sunday as a Sabbath. The opposite is rather true that they thought of them as very distinct days. Sunday worship is it’s own thing and is an important symbol of Christian identity. A more detailed treatment of Sunday worship can be found here. It is worth the read as it is far more comprehensive than what I have mentioned here. Also more links to resources on the matter are provided there.