Gemara

R. Shemen bar Aba brought R. Yochanan's shoes to him to wear. He [first] gave to him the right shoe (R. Yochanan always donned the left shoe first.)

R. Yochanan: You make it appear that it (my right foot) is wounded [hence I should put a shoe only on it]!

R. Yochanan is consistent with what he taught elsewhere;

(R. Yochanan): Just like we put Tefilin on the left hand, we put the left shoe on first.

Question (Beraisa): One puts on the right shoe before the left.

(Rav Yosef): Since the Beraisa and R. Yochanan say differently, one may conduct like either of them.

Objection (Abaye): Perhaps they argue. R. Yochanan never heard the Beraisa, but had he heard it, he would have retracted!

Alternatively, perhaps he heard it, but he knew that the Halachah does not follow it!

(Rav Nachman bar Yitzchak): One who fears Shamayim will fulfill both opinions. Mar brei d'Ravina is such a person.

To fulfill both opinions, one puts on the right shoe, then the left, then ties the left, and then ties the right.

Rav Ashi: I saw that Rav Kahana was not particular [about which shoe he put on first].

(Beraisa): One puts on the right shoe before the left. When removing them, he removes the left shoe first.

One washes or anoints the right [hand] before the left;

When washing the whole body, one should wash the head first, for it is the king over all the limbs.

Rishonim

Tosfos (61a DH d'Avad): One may do like R. Yochanan or like the Beraisa, for they do not argue. R. Yochanan teaches that one may put on the left shoe first like Tefilin, even though the right is more important regarding [blood put on] the thumb and big toe, and Chalitzah. The Beraisa teaches that one may put on the right shoe first, even though the left is more important regarding Tefilin. If so, why was R. Yochanan upset that Rav Shemen gave to him the right shoe first, since one may put either on first? The Ri answers that one may do like he wishes, i.e. if he wants to endear the left, due to the Mitzvah of Tefilin, he may do so, and if he wants he may endear the right, due to its importance. R. Yochanan used to endear the left, and Rav Shemen knew (some texts - did not know) this.

Tosfos (61a DH v'Sayem): The Riva says that the left is more important only regarding tying, since the importance of [the left regarding Tefilin] is regarding tying, but not vice-versa. (One may not put on the left first and tie the right first - PF.) Based on this, for shoes that do not tie, one puts on the right shoe first.

Sefer ha'Yirah (towards the beginning, DH v'Tzarich): One puts on the right shoe first, and then the left. He ties the left shoe, and then ties the right, like Chachamim said, that one who fears Shamayim fulfills both the Beraisa and R. Yochanan. Shoes are like Tefilin. Just like Tefilin is [tied] on the left, also shoes [are tied first] on the left.

Poskim

Shulchan Aruch (OC 2:4): One puts on the right shoe, but does not tie it. Afterwards he puts on the left, then ties it, and then ties the right.

Beis Yosef (DH v'Rabbeinu): The Tur wrote like Rav Nachman bar Yitzchak, who said that one who fears Shamayim will fulfill both opinions. The Poskim omitted this law. It seems that they hold that the Halachah follows Rav Yosef, who says that one may follow either opinion. They learn from Rav Ashi, who said that Rav Kahana was not particular. This implies that also Rav Ashi was not particular. Since he is Basra, we rule like him.

Beis Yosef (DH v'Ika): The Tur rules like Rav Nachman bar Yitzchak. According to him, how does one fulfill both opinions? Putting on the right shoe first is unlike R. Yochanan! How does it help to tie the left first? We can say that even R. Yochanan meant only that one ties the left first. The importance of the left regarding Tefilin is only regarding tying. Even if you will say that he holds that one also puts on the left shoe first, since his whole source for the importance of the left is from Tefilin, in which the left is important regarding tying, through tying the left first, we fulfill his opinion. Tosfos (DH v'Sayem) holds like this. Therefore, he says that for shoes that do not tie, one puts on the right shoe first.

Mishnah Berurah (5): The Torah gives precedence to the right regarding thumbs and the big toe, and everything in which we put the right first.

Mishnah Berurah (6): Regarding felt socks, one need not put on the right first. A lefty, who puts Tefilin on what is the right hand of everyone else, also ties the right shoe first.

Rema: Our shoes do not tie. One puts on the right shoe first.

Taz (3): Rav Nachman bar Yitzchak says that one who fears Shamayim fulfills both opinions and shows a precedence to each. He ties the left first, since the importance of [the left regarding] Tefilin is tying. The Tur omitted removing shoes, and [tying] our shoes, even though the Gemara mentions -removing shoes]. It seems that he rules like Rav Yosef, that one may do like he wants, just since one can fulfill both opinions, he should do the ideal Mitzvah. When one cannot fulfill both opinions, he does like he wants. However, the Beis Yosef says unlike the Tur regarding removing and our shoes.

R. Akiva Eiger: In Chulin 89a, we say that in the merit that Avraham said "from a string to a shoe strap (I will not take the spoils of Sedom)", his descendants merited two Mitzvos, Techeles (on Tzitzis) and straps of Tefilin. Emek ha'Melech (Lech Lecha) explains that since we merited Tefilin corresponding to shoe straps, we tie the left shoe first, just like we tie Tefilin on the left hand.

Magen Avraham (4): (The Gemara says that) one washes or anoints the right first. If he washes the whole body, he washes the head first. Darchei Moshe (1) said that he did not see people particular about this. See the Rambam (Hilchos De'os 4:16, who says that one bathes the entire body in [mildly] hot water that does not scald the body, and just the head in [very] hot water that does not scald the body, and then the entire body in warm water...)

Kaf ha'Chayim (8): Perhaps only regarding putting on shoes and removing them, which is done every day, it is an honor to do the more important one first. Nowadays people do not bathe or anoint the entire body every day, only once in a while, so we are not concerned. The Beraisa discusses when they used to constantly wash and anoint. One who is meticulous even nowadays will be blessed.

Gra: The Beraisa is primary. Abaye said 'perhaps he heard the Beraisa, but he knew that the Halachah does not follow it!' He said so merely l'Ravcha d'Milsa (just to show that one could say oppositely, but really, we are not concerned for this possibility). If not, we could say so whenever an Amora is refuted! Tosfos gave another reason. (Damesek Eliezer - i.e. that R. Yochanan holds that the left is more important only regarding tying. Therefore, for our shoes, one puts on the right shoe first.)

Mishnah Berurah (7): Even if one chanced upon the left shoe first, he waits until he gets the right shoe. One washes or anoints the right first. If he anoints the whole body, the head is first, for it is the king over all the limbs.

Kaf ha'Chayim (9): Also when getting dressed, one covers the head first. However, it suffices to put on the small Kipah worn under the hat or turban. One who has no Kipah, only a hat, puts it on before the shirt.

Halichos Shlomo (2:20, in the name of ha'Gaon R. S. Z. Auerbach, Ztz"l): A woman puts on and ties her right shoe before her left.

Shulchan Aruch (5): When one removes his shoes, he removes the left shoe first.

Beis Yosef (DH v'Efshar): Perhaps this is why the Poskim also omitted the Beraisa's law that one removes the left shoe first (because Rav Ashi was not particular, and he is Basra). No one disagrees! Rather, they hold that just like the Beraisa teaches that one may put on the right shoe, and then the left, if he wants, also the Seifa teaches that one may remove the left shoe first, but it is not a Chiyuv. However, the Tur connotes that the Beraisa obligates putting on the right shoe first. If so, one must remove the left shoe first. If so, why did the Poskim omit this? Perhaps he explains that even though the Beraisa obligates putting on the right shoe first, it permits removing the left shoe first. The Beraisa also teaches that one bathes and anoints the right first. Why is this different than removing, for which the left must be first? Rather, it teaches that one may remove the left shoe first. Do not say that just like one must put on the right shoe first, one must remove it first, because putting on and removing are the same. One is the beginning, and one is the end. Whoever puts on a shoe will later remove it. Since one began with the right, there is no insistence if at the end he removes the left first. However, R. Yonah (Sefer ha'Yir'ah) says that one removes the left first. It seems that he explains the Beraisa simply, to teach that one must remove it first.

Mishnah Berurah (8): This is the honor of the right (foot, that one leaves the shoe on it longer).

Kaf ha'Chayim (12): Also when removing other garments, one removes the left first.