What is the point of putting on your right shoe and then your left? (And if you're a righty to tie your left shoe and then your right.)
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The source for it is in the Gemara (Shabbos 61a), and it is codified as halachah in Shulchan Aruch (Orach Chaim 2:4).
The basic idea behind this practice is:
So to accommodate both preferences, we start with the right side when it comes to putting on items of clothing (it's not only shoes - Mishnah Berurah 2:4; it also applies when bathing - ibid. 2:7), and with the left when it comes to tying.
It also reminds us that even the little things can be done in a Torah fashion. It is reminiscent of the halakha that states a Jew must even give up his life rather than tie his shoes in the non-Jewish manner, in a time of acute persecution.