Being an old timer, I've used both types of diapers and much prefer cloth diapers - except for traveling. I don't know where you would buy those kind of diapers today. They were about the size of 'flour sack' dish towels (another favorite) but a bit thicker and more absorbant. Hey, it's been 40 years since I've bought any diapers.
I also gifted my daughter with a diaper service after the birth of my first grandchild. For some reason, I was not satisfied with them. And that DID surprise me. The diaper service product was more of a liner inside a protective snap-on pant. I just remember my granddaughter having lose stools and the diaper liner did not hold the mess in at all. Three diapers in less than three minutes there. Most of the time, however, they worked very well.
Details on how to use them? I go back to safety pinning them on both sides. There are cute baby safety pins that really are safe. I'd fold the length into three layers - sides to middles - and fold down one end to fit - the front for my boy and the back for my girl - then pin. Over those go plastic pants.
I had a covered wastepaper container that I kept filled with a mixture of bleach, soap and water to hold the used clothes until wash time which was at least every other day. The soiled diapers I rinsed out in the toilet to remove the pooh before putting in the container. I washed mostly baby clothes in one load with a gentle detergent. . . probably not necessary, but I bought the spiel, and no softeners or cleaning agents with perfumes. Then I hung the clean diapers on the line - I still hang out my clothes, nothing like the smell of line hung clothes.
I bought and used only one size of diaper for both my children. The folding method was how size was regulated. 'Course none of mine were premies or even very small at birth.
Be aware that day care providers will most likely require plastic diapers. Not a problem for SAHMs, but something to be considered. Still, you can choose to use one at home and one away. Certainly for any type of traveling, the plastic diaper is the easiest and cleanest way to go so keep a few on hand even if you choose to go cloth.