Praise be to Allah.
When to start wiping over socks
With regard to the beginning of the period when one may wipe over the slippers or socks , it begins from the first time one wipes over them after breaking one’s wudu, not from the moment one puts them on.
How to wipe over socks in wudu
The way in which one wipes over them is to put one’s wet fingers on the toes then pass them over the feet to the shin; the right foot should be wiped with the right hand and the left foot with the left hand. The fingers should be spread when wiping and it should not be repeated. (See al-Mulakhkhas al-Fiqhi by al-Fawzan, 1/43).
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
“This means that what should be wiped is the top of the sock, passing one's fingers from the toes to the shin only, and one should wipe with both hands over both feet together, i.e., the right hand should wipe the right foot and the left hand should wipe the left foot at the same time, just as one wipes the ears, because this is the apparent meaning of the Sunnah, as al-Mughirah ibn Shu’bah said, ‘Then he wiped them both.’ He did not say that he started with the right, rather that he ‘wiped them both’. So this is the apparent meaning of the Sunnah.
If we assume that he cannot use one of his hands, then he should start with the right before the left. Many people wipe the right foot with both hands then the left foot with both hands, but there is no basis for this as far as I know. No matter how it is done, wiping the top of the slipper (or sock) is sufficient but what we have said here is best.” (See Fatawa al-Marah al-Muslimah, vol. 1, p. 250)
You should not wipe the sides or back of the sock because there is no report concerning that.
Shaykh Ibn ‘Uthaymin said: “Someone may say that it is the bottom of the slipper or sock that should be wiped, because this is what is in contact with dust and dirt. But if we think about it we will see that wiping the top of the sock is what makes sense, because this wiping is not for the purpose of cleaning, rather it is an act of worship; if we were to wipe the bottom of the sock that would make it more dirty.” And Allah knows best. (See al-Sharh al-Mumti’ by Ibn ‘Uthaymin, vol. 1, p. 213)
And Allah knows best.