The core obligation of Shnayim Mikra ve-Echad Targum is set out plainly: the Mishneh Torah (Hilchot Tefillah 13:25) and the Shulchan Arukh (Orach Chaim 285:1-2) both require that even though one hears the entire weekly portion read publicly in shul, one must still read it privately twice in the Hebrew and once in the Aramaic targum, down to a verse like 'Atarot v'Divon' that is composed only of place names. Berakhot 8b (8b:1) gives the source and the reward, quoting Rabbi Yehoshua ben Levi's instruction to his sons to complete their portions with the congregation, and promises length of days and years to one who does so, a point also developed by Tiferet Yisrael (13:7).
On whether Rashi's commentary may substitute for the Aramaic targum, the Shulchan Arukh (Orach Chaim 285:1-2) states that studying the portion with Rashi's commentary carries the status of targum, though the God-fearing reads both targum and Rashi; Rav Mordechai Eliyahu (Shut HaRav HaRashi 19:2) and Rav Mordechai Eliyahu (Maamar Mordechai Hilchot Shabbat 2:41) likewise hold that on the plain reading Rashi may be read in place of Onkelos, while according to the kabbalistic view targum itself is required, so the stringent reader includes both. Rav Eliezer Melamed (Peninei Halakhah, Shabbat 5:9) situates this development historically, explaining that once Aramaic was no longer understood, the question arose whether a vernacular translation or Rashi's commentary could substitute for Onkelos. Avnei Yashfeh (8:65) likewise favors Rashi over targum generally, while advising that a verse whose grammatical point in Rashi is not understood be read three times.
As to the mechanics of the reading, Nishmat Shabbat (vol. 2:311) records two valid practices—reading verse by verse or section by section—citing authorities for each and accepting a compromise of reading a section once and then each verse with its targum; Rav Eliezer Melamed (Peninei Halakhah, Shabbat 5:10) confirms that both the Arizal's verse-by-verse custom and the Shelah/Vilna Gaon's section-by-section custom are acceptable, and that completing the reading at any point during the week fulfills the obligation. Rav Mordechai Eliyahu (Maamar Mordechai Hilchot Shabbat 2:2) explains the purpose of the reading as ensuring that a person knows the mitzvot written in the Torah through the study instituted by the Sages, and Rav Mordechai Eliyahu (Maamar Mordechai Hilchot Shabbat 2:43) recommends adding a further commentary each year, such as the Ibn Ezra or the Sforno, alongside Rashi and targum. Avnei Yashfeh (6:54) raises further procedural questions, including whether one must stand for the Aseret HaDibrot during this reading and whether silent reading fulfills the obligation.
Rav Mordechai Eliyahu (Shut HaRav HaRashi 21:3) distinguishes two separate obligations: reading Shnayim Mikra ve-Echad Targum as codified in the Shulchan Arukh, and hearing the communal Torah reading, noting that the former may be done privately from a chumash with makeup time before or after, unlike the latter.
Several sugyot bear on when and how this private reading may be done relative to the communal Torah reading: the Shulchan Arukh (Orach Chaim 146:2) forbids speaking once the Torah reading has begun but permits quietly reading Shnayim Mikra ve-Echad Targum during the public reading for one who has turned away and signaled he is not listening to the reader; this is grounded in the silence required once the scroll is opened, as taught in Sotah 39a (39a:2) and in the sequencing of reader, blessing, and translator described in Sotah 39b (39b:5). The rules for translating the Torah reading itself—one reader paired with one translator, as opposed to the more lenient practice for the Prophets—are set out in Megillah 21b (21b:1-2), while Megillah 17b (17b:1-3) and Megillah 23b (23b:1-3) discuss related rules of language and of how many verses may be read before pausing for translation, forming the background against which the private practice of Shnayim Mikra developed.
Igros Kodesh (letter 8007) recommends a regular daily regimen that includes studying the weekly Torah portion with Rashi's commentary according to a daily division, alongside Tehillim and Tanya, reflecting a practical daily approach to fulfilling this study.
Rav Moshe Sternbuch (Teshuvos V'hanhagos 3:98) rules that even according to poskim who permit substituting another commentary for targum, one should nevertheless also read the portion an additional two times with the actual targum, so as to fulfill the mitzvah according to all opinions.
Rav Ephraim Greenblatt (Revavos Ephraim 5:216) records the practices of leading authorities in this area: Rabbi Moshe Feinstein read the portion twice in Hebrew and then with Onkelos, Rashi, and the Ramban, while the Chazon Ish read the Hebrew text, then Onkelos, and then the Hebrew text again so that Onkelos would clarify his understanding, and the source laments the broader decline in attention to studying the translations that earlier authorities had emphasized.