For dish rags and cleaning rags, I just cut up old shirts. Cotton t-shirts and worn-out flannel shirts work great. They are not particularly absorbent, but they do the job just fine. These are shirts that have holes or stains and would do no one any good at Goodwill - having a boatload of those shirts during a de-clutter was part of what led me to do away with paper towels. It was mostly "hey...I have these shirts...what do you suppose I could do with them besides throw them in the garbage?" We have many more rags than we use - that'd be more than a load of laundry's worth. (I may be a recovering pack rat who can't say no to a free t-shirt.)
When Squeak outgrew the smallest pre-folds we had, we designated about half of them as "un-paper towels" - i.e. things for cleaning up wet messes. Before that, we were using dish towels, which was less than ideal.
Originally we never had napkins in the house - paper or otherwise. In a pinch, we were giving people the cleaning rags (before the rags, paper towels). So dinner parties and messy toddlers got the same treatment...which grew to feel uncivilized, so I ordered some napkins from a shop on Etsy, and I've been happy with them (not that we host dinner parties too often...).
Storage and laundry: We have a giant shopping bag full of rags under the kitchen sink, and another bag with the outgrown prefolds/unpaper towels. When they are used, they get laid or hung out to dry. When they're dry, they go in a cotton laundry bag. Once a week (during my supposed once-a-week cleaning routine of the whole house) I make sure all the dry rags around the house are in the bag. If the bag is full, I do a load of rag laundry. Otherwise I wait another week. I probably do a load of rags every three weeks, or maybe less often. This load includes dish towels, the unpaper towels/prefolds, and the cut-up shirt cleaning cloths.
In terms of the wash routine, I do it the same way I launder cloth diapers. We have such a ridiculous supply (so many rags that I've actually got some t-shirts stockpiled to be cut up later - anybody want them???), that I've been known to throw away rags when they get super oily (which happens with some frequency because our cast-iron skillet ultimately gets a coat of oil after every use). Other than that, sometimes I give them a long soak in hot water and detergent before the wash or I will do an extra hot rinse or two. I don't deal with trouble-shooting them, but just throw one away if it is feeling gross. Call me wasteful!
What about you? Do you have more "official" cloth rags (sometimes I think a rag with sewn edges in a nice hue would make cleaning the kitchen table several times a day a little less of a chore)? Or do you use paper towels? Do you have any additional laundry tricks, or things that have worked really well? Perhaps a material that is stellar for a particular household task? Perhaps you have a storage method you love, like a towel house?
**I have absolutely no affiliation with the Etsy store I've linked to on this page and am not vouching for their quality. Our napkins were made by them, and we're happy with them, but I am not giving any more or less endorsement than that. I have not, for instance, ever seen one of their towel houses outside a picture on Etsy.