The Shulhan Aruch (Orah Haim 282) rules that minors – youths under the age of Bar Misva – may be called to the Torah, and they count toward the required seven Aliyot on Shabbat. The Arizal (Rav Yishak Luria of Safed, 1534-1572) maintained that a minor may be given only the seventh Aliya on Shabbat, but other authorities allow giving a minor any of the Aliyot. Another minority position is that of the Magen Abraham (Rav Abraham Gombiner, Poland, 1637-1682), who ruled that a minor should not receive the Aliya of Kohen, because granting the first Aliya to a Kohen is required by virtue of the Torah obligation of "Ve’kidashto" – that we must show honor to the Kohanim. The Torah requires giving the Kohanim honor because "Lehem Elokecha Hu Makrib" – they offer sacrifices in the Bet Ha’mikdash. The Magen Abraham therefore rules that since Kohanim who are minors may not offer sacrifices, they are not included in the obligation of "Ve’kidashto," and thus should not receive the first Aliya. Other authorities, however, refute this argument. As Hacham Ovadia Yosef explains (Hazon Ovadia – Shabbat, p. 249), even Kohanim with a physical defect – who are disqualified from offering sacrifices – are given the first Aliya to the Torah, and thus a minor, too, is eligible to receive this honor.
Hacham Ovadia Hedaya (1889-1969), in his Yaskil Abdi, writes that the accepted practice is not to give Aliyot to minors, but Hacham Ovadia observes that even if this was true in Hacham Ovadia Hedaya’s synagogues, the accepted practice is to allow minors to receive Aliyot.
Some Halachic authorities, including Hacham Ovadia Hedaya and Hacham Bension Uziel (1880-1953), ruled that a minor may not read the Torah, just as a minor may not serve as Hazan, or read the Megilla on Purim. Since he is not yet obligated in Misvot, he is unable to fulfill the Misva on the congregation’s behalf. Hacham Ovadia Yosef, however, disagrees, and allows minors to read the Torah. He notes the comment of the Meiri (Rav Menahem Meiri, 1249-1315) that there is no personal obligation to read or hear the Torah, like there is to pray and to read the Megilla. The obligation of Torah reading is that the Torah should be read in the synagogue, not that every individual must read or hear the reading. Therefore, even a minor, who is not yet formally obligated in Misvot, may perform the reading. Hence, a minor may receive an Aliya and may conduct the Torah reading.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) cites the ruling of the Peri Megadim (Rav Yosef Teomim, 1727-1792) that if no adult capable of reading the Torah is present in the synagogue, and there is a minor who is able to read, the minor may read the entire Parasha.
Hacham Ovadia Yosef demonstrates from many Rishonim (including the Rambam, Rabbenu Yeshaya Rishon, Mahram Me’utenberg, Abudarham, and Tashbetz) that this Halacha, allowing minors to receive Aliyot, applies not only on Shabbat, but also on Mondays and Thursdays. This ruling is very relevant nowadays, when many people have the custom to hold a Bar Misva celebration sometime before the boy’s thirteenth birthday, on a Monday or Thursday. The boy may read the Torah on the day of the celebration, despite the fact that he has not yet actually become a Bar Misva.
The Aruch Ha’shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) cites Masechet Sofrim as establishing that there is a Misva for women to hear the Torah reading. He explains, however, that Masechet Sofrim speaks not of a technical Halachic obligation, but rather of a stringency that is worthwhile for women to adopt when possible. Clearly, women are not required to hear the Torah reading, with the possible exception of the reading of "Zachor" on the Shabbat before Purim, when women should make an effort to come to the synagogue to hear the reading.
Summary: Minors may receive any Aliya, and may even read the Torah. If there is no adult present who can read, a minor may even read the entire Parasha. This applies both on Shabbat and on Mondays and Thursdays.