Our Rabbis taught: The place [in the Torah] where they leave off in the morning service on Sabbath is the place where they begin at Minhah; the place where they leave off at Minhah [on Sabbath] is the place where they begin on Monday; the place where they leave off on Monday is the place where they begin on Thursday; the place where they leave off on Thursday is the place where they begin on the next Sabbath. This is the ruling of R. Meir. R. Judah, however, says that the place where they leave off in the morning service on Sabbath is the place where they begin on [Sabbath] Minah, on Monday, on Thursday, and on the next Sabbath. (b. Meg. 31b)
Most scholars believe that ancient Church reading practices of the Bible are sourced in the Jewish Synagogue. The above quotes gives insight into the practice of reading through the Hebrew Bible. This may be to ancient roots of the Christian liturgy.
Our Rabbis taught: [The one who reads] opens the scroll and sees [the place], then rolls it together and says the blessing, then opens it again and reads. So R. Meir. R. Judah says: He opens and looks and says the blessing, and reads. What is R. Meir's reason? — It is similar to that of ‘Ulla [in a parallel case]; for ‘Ulla said: Why did they lay down that he who reads from the Torah should not prompt the translator? So that people should not say that the translation is written in the Torah. So here [R. Meir's reason is], so that they should not say that the blessings are written in the Torah. And [what says] R. Judah [to this]? — With regard to translation a mistake might be made, but no mistake will be made with regard to the blessings. (b. Meg. 32a)
I am no Talmud scholar, so I am not exactly sure what is going on here. Either the reader is immediately translating the Hebrew into the vernacular, so that the reader looks at the scroll and reads the passage in Hebrew, looks away, closes the scroll and translates into the vernacular; or the reader is reciting some blessing that is not written in the Tanakh and does the same. Either way, there were strict reading practices and habits to insure the listeners were never confused as to what was being read to them.
R. Shefatiah said in the name of R. Johanan: When one rolls up a scroll of the Torah, he should make it close at a seam. R. Shefatiah further said in the name of R. Johanan: One who rolls together a Sefer Torah should roll it from without and should not roll it from within, and when he fastens it he should fasten it from within and should not fasten it from without. (b. Meg. 32a)
Not directly related to reading, but I find it interesting the great care with which their precious scrolls were treated at all times. Here in their Mishnaic teachings are directives on how to roll up the scroll, outside-in and not inside-out.
R. Shefatiah further said in the name of R. Johanan: If one reads the Scripture without a melody or repeats the Mishnah without a tune, of him the Scripture Says, Wherefore I gave them also statutes that were not good, etc. Abaye strongly demurred to this, saying, Because he cannot sing agreeably, are you to apply to him the verse,‘ordinances whereby they shall not live’? No; this verse is to be applied as by R. Mesharshia, who said: If two scholars live in the same town and do not treat one another's Halachic pronouncements respectfully, of them the verse says, I gave them also statutes that were not good and ordinances whereby they should not live. (b. Meg. 32a)
A very interesting insight into the method of reading, it was in a cantillated tone. Not monotone and conversational like we read today (see Gamble, 225-226). Note especially how the manner of reading is associated with singing.
R. Parnak said in the name of R. Johanan: Whoever takes hold of a scroll of the Torah without a covering is buried without a covering. Without a covering, think you? — Say rather, without the covering protection of religious performances. Without religious performances, think you? — No, said Abaye; he is buried without the covering protection of that religious performance. (b. Meg. 32a)
Again, not specifically related to reading, but notice how the scrolls were to be kept in some type of covering at all times.
R. Jannai the son of the old R. Jannai said in the name of the great R. Jannai: It is better that the covering [of the scroll] should be rolled up [with the scroll] and not that the scroll of the Torah should be rolled up [inside the covering]. And Moses declared unto the children of Israel the appointed seasons of the Lord. It is part of their observance that [the section relating to] each one of them should be read in its season. (b. Meg. 32a)
Again the mention of ensuring the scroll is covered. But also note that there is a "season" by which certain sections of the Tanakh are read. Again, this practice may be the source of the ancient Christian liturgy.
The Babylonian Talmud. Edited by I. Epstein. 35 vols. London: Soncino Press, 1938-52.
Gamble, Harry Y. Books and Readers in the Early Church: A History of Early Christian Texts. New Haven: Yale University Press, 1995.