Is there any special order that should be followed when bathing the body?
Yes. When bathing one’s body one begins with washing his head, as the head is the king of all the limbs. One then washes his right hand first and only then proceeds to his left.
Explanation: The law which requires one to precede his right side over his left side when getting dressed and undressed, likewise applies when bathing, although when bathing we give precedence to the head being that it is the king of all limbs.
Sources: Admur Kama 2:6, Basra 2:4; M”A 2:4; Ben Ish Chaiy Vayishlach 17; In the Basra ibid Admur writes to wash the right hand first. However in Kama ibid the word hand is omitted and perhaps it refers to the entire right side of the body; that it is to be washed first; Other opinions: The Aruch Hashulchan 260:3 writes that after washing the head one is to wash the heart, right hand, left hand, right leg, left leg. The same applies regarding the order of applying soap. The M”A ibid brings the Darkei Moshe which writes that he has not seen people being careful in the order of bathing and smearing. The Kaf Hachaim 2:8 brings the Mateh Yehuda 2:5 which suggests that perhaps the reason for this is because back then it was not common to bathe every day and hence it was not considered a matter of respect to precede one limb over the other. Nevertheless he concludes that one who is careful in this is blessed.