The recommendation to use the right first in things that entail honour, like wudu', ghusl, and tayammum

Riyad as-Salihin (The Meadows of the Righteous)

by Imam Nawawi

99. Chapter: The recommendation to use the right first in things that entail honour, like wudu', ghusl, and tayammum, putting on clothes, shoes, leather socks and trousers, entering the mosque, using the siwak stick, using kohl, cutting nails, trimming the moustache, plucking the armpit and shaving the head, giving the salam in the prayer, eating and drinking, shaking hands, greeting the Black Stone, leaving the lavatory, taking the gift, and other such things. The recommendation to use the left in the opposite of that, like blowing the nose, spitting, entering the lavatory and leaving the mosque, removing leather socks, sandals, trousers and clothes, cleaning oneself in the lavatory, doing unclean things, and the like

Allah Almighty says, "As for him who is given his Book in his right hand, he will say, 'Here, come and read my Book!'" (W69:18; H69:19)

and the Almighty says, "The Companions of the Right. What of the Companions of the Right? The Companions of the Left. What of the Companions of the Left?" (W56:8-11; H56:8-9)

721. 'A'isha reported that the Prophet, may Allah bless him and grant him peace, liked to begin with the right in all things - in purification, combing his hair and putting on his shoes. [Agreed upon]

722. 'A'isha said, "The Messenger of Allah, may Allah bless him and grant him peace, used his right hand for purification and eating, and his left for the lavatory and whatever was offensive." [Abu Dawud]

723. Umm 'Atiya reported that the Prophet, may Allah bless him and grant him peace, told the women when they were washing his daughter Zaynab, "Begin with her right hand side and the places which are washed in wudu'." [Agreed upon]

724. Abu Hurayra reported that the Messenger of Allah, may Allah bless him and grant him peace, said, "When one of you puts on sandals, he should start with the right foot and when he removes them, he should start with the left foot so that the first sandal he puts on and the last he takes off is the right one." [Agreed upon]

725. Hafsa reported that the Messenger of Allah, may Allah bless him and grant him peace, used to use his right hand for food, drink and clothes, and he would use his left hand for other things. [Abu Dawud and at-Tirmidhi]

726. Abu Hurayra reported that the Messenger of Allah, may Allah bless him and grant him peace, said, "When you dress and when you do wudu', begin with your right." [Abu Dawud and at-Tirmidhi]

727. Anas reported that the Messenger of Allah, may Allah bless him and grant him peace, arrived in Mina and went to the Jamra and stoned it. Then he went to his place in Mina and sacrificed. Then he said to the barber, 'Take it off,' indicating his right side and then his left. Then he began to give it [his hair] to the people." [Agreed upon]

In one variant, "When he had stoned the Jamra and completed his sacrifice and wanted to shave his head, he indicated his right side to the barber and had it shaved. Then he called Abu Talha al-Ansari and gave it [his hair] to him. Then he indicated the left side and said, 'Shave.' He had it shaved and gave it [his hair] to Abu Talha and said, 'Divide it between the people.'"