Nowadays, there are products to purchase for every every possibility, occupation and hobby that would make the need for humble rags obsolete. But I still prefer cotton rags for their feel and their cost. They are an old-fashioned method of recycling that makes a lot of sense to me.
The most absorbent rags are made from worn out or stained towels, worn out cotton athletic socks and worn out or stained cotton t-shirts. Other cotton clothes like worn out flannel pajamas work too. Worn out cotton sheets have other uses, like becoming drop cloths for painting, but don't make very good rags.
To make sock rags, cut or rip socks open on the bottom where they are most worn so the unworn top surface becomes most available to use. For t-shirts, cut off and discard the sleeves, neck, hem and seams, then cut or tear the body into the size rags you want, usually 4 to 6 pieces. Also, cut off any screen printing so all the surfaces are soft and absorbent. Towel rags are nice cut into approximately the size of hand towels.
Keep rags handy in the kitchen, bathroom and shop. Sock and towel rags are very useful for all kinds of cleaning jobs, for spills and messes, and for any job that may stain, like wiping up muddy footsteps or cleaning food off little children's hands and faces. T-shirt rags are nearly lint-free so work great for applying furniture polish, etc. We use t-shirt rags to dry our eyeglasses each morning and sock rags to wash our cars. In all these cases, the rags are rinsed and hung in the laundry room to dry after use, then washed and dried each week along with our bath towels.
Rags are also unmatched for any job like waxing your car, painting, furniture refinishing or working on greasy machines where you will need to throw them away afterword.