if anyone wants to easily see this inside it is in kitzur shulchan aruch in halachos of erev shabbos chapter ayin Beis (72) paragraph Yud daled (14)right hand cutting order = 2,4,1,3,5 left hand = 4,2,5,3,1
and as written above, most poskim say begin with left hand.
In the book Halachichaly speaking vol.1 there lists separate sources for nail cutting and their order that are interesting such as long nails might be a sign of a thief or that Torah learning may be forgotten to others that it shows it is a sign of lack of derech eretz