It is Sunnah to begin with right side in every action which is connected with honor.
Our beloved Nabi Muhammad (saw) always put on his right shoe first when he wore the shoes. The right shoe was also the last to be taken off his foot.
Our beloved Nabi Muhammad (saw) was reported to have said that :
إِذَا انْتَعَلَ أَحَدُكُمْ فَلْيَبْدَأْ بِالْيَمِينِ، وَإِذَا نَزَعَ فَلْيَبْدَأْ بِالشِّمَالِ، لِتَكُنِ الْيُمْنَى أَوَّلَهُمَا تُنْعَلُ وَآخِرَهُمَا تُنْزَعُ
“If you want to put on your shoes, put on the right shoe first; and if you want to take them off, take the left one first. Let the right shoe be the first to be put on and the last to be taken off” (Bukhari&Muslim)