Regarding a prescribed order of readings for Jews: there is no one prescribed order. The length of a "reading cycle" has varied over time: as the article on the "Triennial Cycle" in the Encyclopedia Judaica (2nd edition) notes, "In traditional synagogues, the Pentateuch is read in one year. Reform Judaism (and some Conservative synagogues) has, however, reverted to the ancient Palestinian custom of a triennial cycle" (vol. 20, p. 143). Another article in the Encyclopedia Judaica, 2nd edition, ("Torah, Reading Of") goes into a bit more detail about how the Torah is read today and includes a number of tables of readings. There is also an article on the "Triennial Cycle" in the first edition of the Encyclopedia Judaica; it's available for free online (http://www.jewishencyclopedia.com/view.jsp?artid=327&letter=T&search=triennial).

Tables of readings for the Jewish faith can be found online -- for example, http://www.mechon-mamre.org/p/pt/readingp.htm has a table for the annual reading of the Torah, along with another table of readings for special Sabbaths and holidays. Note that the Torah reading is accompanied by readings from the Prophets (called Haftaroth; the article mentioned above, entitled "Torah, Reading Of," notes that "[t]he practice of "completing" the Torah reading with a passage from one of the prophetic books, the haftarah ("completion"), is mentioned in the Mishnah (Meg. 4:1–2); the origins of the custom, however, are obscure" (vol. 20, p. 50). These tables are keyed to a calendar (http://www.mechon-mamre.org/jewfaq/current.htm); if you start with the calendar, you can get to the correct reading for a particular day by clicking the links on the calendar.

Online orders of reading for the triennial cycle are available, as this example: http://www.hebcal.com/hebcal/; to see weekly readings, make sure you check the "Weekly sedrot on Saturdays" option under the "Include events" section on the left of the page. The calendar will have links to the readings for both the annual and the triennial cycles.

Regarding a prescribed order of reading for Muslims: the Encyclopedia of the Qu'ran notes the following about recitation of the text: "During the fasting month of Ramaḍān, the entire Qur'ān is read over the course of the month in night prayers called tarāwīḥ. One of the standard divisions of the Qur'ān is its partition into thirty equal, consecutive parts, or juz' (pl. ajzā'); this sectioning facilitates complete recitation over the course of a month. In addition, during Ramaḍān or during the days of the pilgrimage, pious Muslims may recite the entire Qur'ān in one night. Muslims read the Qur'ān frequently as an act of supererogatory piety, and recitation — especially at night — is performed by committed Muslims" (from the article "Recitation of the Qur'ān" by Anna M. Gade, in Encyclopaedia of the Qur'ān, gen. ed. Jane Dammen McAuliffe (Washington DC. Brill, 2009). This article goes on to mention some of the sūras that are frequently evoked in specific situations, with emphasis on the correct recitation (including pronunciation) of the Arabic. As best we know, there is not a "cycle" or "order of readings for the Qur'ān, other than the Ramadan division of readings.