I learned 3 different ways - but they are all similar.
Mom had an egg separator that worked pretty well. She placed it over a measuring cup, broke the egg into it, poured the yolk into one bowl and the white into another.
Grandma used 4 bowls and her fingers - she broke the egg into one bowl, poured it into her hand over the second bowl, dropped the yolk from her fingers into the yolk bowl and poured the white into the whites bowl.
My other grandmother did the shell-to-shell thing - but again separating each egg over a 3rd bowl.
I, in my infinite wisdom, decided I could skip a step and save the time and trouble of having to wash that 3rd bowl. Yep - I cracked the egg, did the shell-to-shell thing over my bowl of whites like I had several times before ... and then one fateful day the yolk of the last egg broke! My batch of whites was contaminated, ruined! IF I had separated it into a 3rd bowl first I could have just reached for another egg, and another bowl, and gone on. So, the 30-seconds I might have saved having to wash another bowl or two turned into having to stop what I was doing, change clothes, walk about 1/2 mile to the the store (DW had the car) and get another dozen eggs, come home and start all over!
No matter how you separate your eggs (separator, shell-to-shell, or with your fingers) use a 3rd bowl to isolate each egg so you don't contaminate the whole batch if you have a yolk break. And, it's also much easier to fish out a piece of broken shell from just one egg than a bowl of several.