The Messenger of Allah liked to start on the right when purifying himself, when combing his hair and when putting on his footwear. (Using translation from Ibn Mājah 401)   

كَانَ رَسُولُ اللهِ ﷺ يحِبُّ التَّيَمُّنَ فِي طُهُورِهِ إِذَا تَطَهَّرَ وَفِي تَرَجُّلِهِ إِذَا تَرَجَّلَ وَفِي انْتِعَالِهِ إِذَا انْتَعَلَ  

[Machine] The Prophet ﷺ used to prefer to do everything on his right side to the best of his ability, including walking, wearing shoes, and performing ablution.  

أَنَّ النَّبِيَّ ﷺ كَانَ يُعْجِبُهُ التَّيَمُّنُ مَا اسْتَطَاعَ فِي شَأْنِهِ كُلِّهِ فِي تَرَجُّلِهِ وَتَنَعُّلِهِ وَوُضُوئِهِ  

The Prophet ﷺ said: When you put on (a garment) and when you perform ablution, you should begin with your right side. (Using translation from Abū Dāʾūd 4141)   

قَالَ رَسُولُ اللهِ ﷺ إِذَا لَبِسْتُمْ وَإِذَا تَوَضَّأْتُمْ فَابْدَءُوا بِأَيَامِنِكُمْ