In 1 Timothy 4:13, Paul explicitly tells Timothy to read the Scripture publically in the early church:
Until I come, devote yourself to the public reading of Scripture, to preaching and to teaching.
We see elsewhere that he did not want the reading limited to the Scriptures of the Old Testament, but those connected with the dispensation of Christ as well (thereby putting those writings on par with the inspired texts of the Old Testament):
Colossians 4:16 And when this letter has been read among you, have it also read in the church of the Laodiceans; and see that you also read the letter from Laodicea.
1 Thessalonians 5:27 I put you under oath before the Lord to have this letter read to all the brothers.
The Apostle John likewise assumes that the book of Revelation will be read in the churches:
Revelation 1:3 Blessed is the one who reads aloud the words of this prophecy, and blessed are those who hear, and who keep what is written in it, for the time is near.
We then see this patten continued into the liturgy of the primitive church.* Second Clement, an early Christian sermon, says, “So then, brothers and sisters, after the God of truth I am reading to you this appeal to pay attention to the things that have been written in order that you may save yourselves and also the one who is reading among you” (19.1).
Justin Martyr in the 150’s said that on Sundays, “the memoirs of the apostles or the writings of the prophets are read, as long as time permits.” (1 Apology, 67).
The late 2d century Christian theologian Tertullian said, “We meet together in order to read the sacred texts.” He continues, “With the holy words we feed our faith, we arouse our hope, we confirm our confidence” (Apology 39). In another place he stresses why the public Scripture reading is so important: “The church unites the Law and the Prophets in one volume with the writings of evangelists and apostles, from which she drinks in her faith.”**
Clement of Alexandria (c. 182-202) talks of early Christians in their worship, “always giving thanks in all things to God through righteous hearing and divine reading.” (Miscellanies 184.108.40.206)
Later, Theodore of Mopsuestia in the late 4th century said, “All of us, having come to faith in Christ the Lord from the nations, received the Scriptures from them and now enjoy them, reading them aloud in the churches and keeping them at home.” (Commentary on the Twelve Prophets***)
So it appears evident that throughout the first five centuries of the church, the Scriptures were, in accordance with the command of Paul, regularly read in the church’s public worship.
**David Bercot, Dictionary of Early Christian Beliefs, (Hendrickson, 1998), 606
***Fathers of the Church 108; trans. Richard C. Hill; Washington, D. C.: Catholic University Press, 2004, 289.