Gemara

29a (Beraisa): A man is obligated to teach a trade to his son;

R. Yehudah says, anyone who does not teach his son a trade, (it is as if) he teaches him robbery.

82a (Mishnah - R. Meir): One should teach his son a clean, easy trade and pray to Hash-m, for wealth or poverty do not result from the trade, rather, from merit.

Aba Guryan of Tzadyin, citing Aba Gurya: One should not train his son to drive donkeys, camels, wagons or ships, or be a shepherd or grocer, for they steal.

R. Nehurai: I will not teach my son any trade, only Torah, for one enjoys its reward in this world, and the principal remains intact for the next world.

Trades involving women lead to bad conduct, such as jewelry makers, flax combers, people who fix grinders, peddlers, weavers, barbers, launderers, bloodletters, bathhouse attendants, and tanners.

(Bar Kapara): A man should teach his son a clean, easy trade.

(Rav Yehudah): This is embroidery.

(Beraisa - R. Meir): A man should teach his son a clean, easy trade and pray to the Source of wealth. Wealth or poverty are not from the trade, rather from Hash-m - "mine is the silver and gold."

(Beraisa - R. Nehurai: I will not teach my son any trade, only Torah. Trades help one only in his youth. When he is old, he will starve. Torah helps one in his youth - "those who hope to Hash-m will get renewed strength..." It gives him a good end in his old age - "they will still flourish when old, they will be fresh".

Makos 8b: Why does a father go to Galus (a refuge city) for killing his son? It is always a Mitzvah for him to hit him (so he will learn)!

Answer: The case is, he was teaching him carpentry when he killed him.

Question: It is also a Mitzvah to teach one's son a trade!

Answer: The case is, the son already knew a trade.

Rishonim

Rif and Rosh (12a and 1:40): One must teach a trade to his son. R. Yehudah says, if one does not do so, it is as if he teaches him robbery.

Rambam (Hilchos De'os 6:10): One must be very careful not to pain orphans... if one afflicted them in order to teach them a trade, it is permitted.

Rambam (Hilchos Shabbos 24:5): On Shabbos, one may arrange to hire someone to teach a trade to his son... this is a Mitzvah. Your desires are forbidden. Shamayim's desires are permitted.

Rambam (Hilchos Rotze'ach 5:5): If one killed his son b'Shogeg while teaching him another trade that he does not need, he goes to Galus. If he was teaching him a trade (that he needs), he is exempt.

Perush ha'Mishnayos (Sof Kidushin): One is obligated to teach to his son a clean trade. Therefore, he may not teach him to teach (children)...

Einayim l'Mishpat (Kidushin 29a 8 and 82b): In the Mishnah Torah, the Rambam brings a Mitzvah to teach to him a trade, but we do not find that it is an obligation. It is difficult to say that R. Nehurai holds that there is no Mitzvah and we rule like him. R. Shimon ben Elazar holds that one need not teach him a trade, for everything depends on one's deeds.

Igros Moshe (OC 2:111 DH b'Davar): Income is decided on Rosh Hashanah, but Hash-m decreed that people will receive it only through work or business - "b'Ze'as Apecha Tochal Lechem." R. Nehurai agrees that one may not rely on miracles. However, he holds that most children cannot properly learn Torah and a trade at the same time. Since the Torah commands the father to teach Torah to his son, he is forced to trust that Hash-m will present his son with work when the son matures and needs to support his family. The Rambam obligates a man to work, but regarding children he rules like R. Nehurai. We do not find that he obligates teaching a trade to one's son. Just like Hash-m's decree does not mandate working on Shabbos, it does not mandate neglecting to teach children Torah properly. The Rambam mentioned teaching a trade only in passing. He refers to a child too weak to learn Torah, or to someone older.

Poskim

Shulchan Aruch (OC 306:6): On Shabbos, one may discuss Shamayim's desires, e.g. to arrange to hire someone to teach a trade to his son. One may discuss if he wants to be hired, but one may not hire him or mention the wage.

Magen Avraham (13): This is a Mitzvah, for if he has no trade, he will steal.

Beis Yosef (DH veha'Rav): Maharam, the Ramban, the Ran and the Magid Mishneh forbid mentioning the wage. The Mordechai permits if it is needed.

Bi'ur Halachah (DH u'Lelamdo): Tana'im argue about this. Some say that one must teach his son Torah and a trade or business. R. Nehurai says that he teaches only Torah. All agree that one must teach to him Torah and Yir'as Shamayim before and while teaching him a trade, for without this he will transgress in his trade many kinds of theft and breaches of Torah. Learning only Torah causes sin, all the more so learning only a trade!

Kaf ha'Chayim (51): One must ensure that his son learns from a craftsman with Yir'as Shamayim, lest he learn from his ways.

Rema (YD 246:21): If one learns Torah without working, this causes sin and in the end he will steal.

Gra (60): We learn from R. Yehudah.

Aruch ha'Shulchan (OC 156:1): A trade is a good thing. Many err and say that it is disgraceful. One is obligated to teach his son a trade. Many Tana'im and Amora'im had trades. The Medrash (Bereishis Rabah 74) says that Melachah is dearer than Zechus Avos. Zechus Avos rescues money. Yakov told Lavan "If not for Elokei Avi Avraham... you would have sent me away empty." Melachah saves life - "...v'Es Yegi'a Kapai Ro'oh Elokim v'Yochach Emesh" (and stopped Lavan from killing Yakov).