Moshe Rabaynu decreed that there should be public Torah readings every Shabbos, Monday and Thursday. L’chatchila it should be read in the morning but b’dieved the entire day is the time. (ShA 1 – MB intro, 1)
Ezra decreed an additional reading at Mincha of Shabbos to lure the loiterers into shul. (MB 1ntro, ShaH 1)
The readings are at least 10 pesukim and are divided up for 3 readers and no more. There should not be any haftaras either. (ShA 1 – MB intro)
Our minhag is to limit the amount of readers to 3, even if there are 2 chassanim in shul and both are yisroelim. (MB 3)
The end of the kriah of Shabbos morning marks the beginning of the kriah of Shabbos Mincha, the upcoming Monday and Thursday, and the next Shabbos morning. (ShA 2)
A cohen reads the Torah first, then a levi and after him a yisroel. (ShA 3)
A cohen should be first to read the Torah, to bentch, and to be given the first portion of food. He may give away the honor to someone greater than himself, except when it comes to being the first to read the Torah. Chazal decreed that a cohen and a levi may not forgo on this. (MB 9)
If there is no cohen present, give the first chance to read to the greatest person in wisdom and number (of students?) even if there may be an older man present. (MB 9)
If the only cohen in shul does not know how to read, you should still call him up even before a gadol b’Yisroel provided that the cohen is able to at least say the words along with the baal koray. You do not need to test the cohen first – assume he can. (ShA 4 – MB 15)
In the following cases call up a yisroel to read first and not the cohen:
The cohen is in the middle of the berachos of Krias Shema.
The krias haTorah is for the optional fasts after Pesach and Sukkos (B’HaB) and the cohen is not fasting. In this case, it is best that the cohen exits the shul. (ShA 5)
Do not make the tzibbur wait for the cohen to finish his tefilla. However, if there is no other cohen and he is between the paragraphs of Birchos Krias Shema, he should be called up l’chatchila before a yisroel. This applies, as well, if the cohen is saying Pesukay D’zimra. (MB 16,17)
In the following cases let the yisroel finish his reading, he will count as one of the required number of readers and do not call up the cohen after him:
The cohen entered the shul only after the yisroel began his preceding beracha “Boruch atta Hashem”.
A mistake was made and they thought the cohen was not present. (ShA 6 – MB 19,20)
If the cohen entered before the yisroel said “Boruch atta Hashem”, the yisroel should just move to the side, let the cohen and levi take their turns, and then the same yisroel should read after them. That the yisroel already announced “Borchu …” and the k’hal responded “Boruch Hashem ham’vorach…” is not considered l’vattalla. Nevertheless, the cohen should repeat “Borchu…”. (ShA 6 – MB 21)
If no cohen is present, call up a yisroel to read first and do not call a levi second (do not call him up except for acharon or maftir). If the yisroel is no greater than the levi you can call the levi first. (ShA 6 – MB 24)
If you ever call a non-cohen first, announce “…bimkom cohen.” (Rama 6)
If you mistakenly thought that there is no levi present to read after the cohen and the cohen began his preceding beracha for the second reading “Boruch atta Hashem”, let the cohen continue with the second reading. (ShA 7 – MB 26)
If there is no levi present, do not allow another cohen to take his place. The same cohen who read the first reading should read the second one. (ShA 8)
Do not call levi after levi. (ShA 9)
The minhag Ashkenaz is not to call up another cohen or levi for the rest of the required alios (7 on Shabbos – 5 on Yom Tov) unless very necessary. For example if a cohen or levi was called up (e.g. for revii) by mistake and it is awkward to have him wait until the end – as long as a yisroel had an aliya in between, the cohen or levi may take his turn. Another example of necessity would be Simchas Torah when there are many cohanim and leviim and everyone gets an aliya – after calling up the usual cohen, levi, and yisroel, do that order again and again until all the cohanim and leviim receive their alios. (Rama 10 – MB 35,37)
Whenever you call a cohen or levi after the first two alios it is important to announce that he is a cohen or levi. (ShA 10 – MB 33)
The Levush writes a further restriction: do not call up a cohen or levi even after the 7 required alios. Only the last additional aliya (before Kaddish) or Maftir may be given to them. Nevertheless, you can be lenient in the case that there are two chassanim who each have cohanim and leviim in their respective retinue. Split the parsha into two. Call up 7 alios for the first half beginning with a cohen and levi. Then begin again 7 aliyos for the second half beginning with a cohen and levi. (MB 36)
The Chassan Torah on Simchas Torah can be a cohen – it is considered like the last aliya. (MB 36)
If a yisroel is called for an aliya and he is not present, you can call up another yisroel by name. But if the same thing happened for a cohen or levi aliyah, do not call up another cohen or levi by name. (ShA 11 – MB 38)
If the cohen or levi is present, just he is in the middle of Shemoneh Esray – you may call another cohen or levi by name. Similarly, if the cohen or levi is not present for an aliyah that even a yisroel can take. (e.g. acharon, maftir) you may call up another cohen or levi by name. (MB 39, 41)
A gabbai to call out the aliyos who is a cohen may call up other cohanim – people will not suspect that the gabbai is not a cohen. (Rama 11 – MB 42)
A cohen ovail who is forbidden to learn Torah is allowed to be called up first on Shabbos, only when there is no other cohen. It is better still, for him to exit the shul beforehand whether on Shabbos or during the week. (MB 45)
It is considered a disgrace for a sefer Torah to be brought to a person who needs kriah unless he is a gadol baTorah. It is an honor for people to go to where the sefer Torah stays. (MB 47 – Rama 14)
The poskim argue about cases where the people that need kriah cannot come to shul (e.g. imprisoned, ill). If there are 10 men who need the kriah it is allowed. Less than 10, you can be lenient only for Parshas Zachor and Parshas Para. (ShA 14 – MB 46,47)
On Rosh Hashana and Yom Kippur you may take a sefer Torah from the shul and read it somewhere else so that more people can get alios. On one condition: sell the alios to raise more money for the shul. (MB48)
It is a kavod haTorah to sell the alios. (ShaH 46)
The way to get around the problem of disgrace in all cases would be to prepare a place for the sefer Torah for a day or two before the kriah or for a day or two after the kriah at the location of the temporary kriah. (Rama 14 – MB 49)
If you will be leaving the sefer Torah in the temporary location for a day or two after the kriah, be sure to first put the sefer Torah in its prepared place and only afterwards take it out to read. Do not take it from shul directly to read. (MB 49)