Having a dedicated set of rags or cloths for cleaning is important.
Random torn up t-shirts, odd socks, or old towels and sheets
will doom your effectiveness before you even begin.
Have enough, 20-30, to tackle the worst of cleaning days and still be able to clean other parts of the house using fresh cloths. Wash them separately, in hot water, from any other laundry because....ew! Store them in a laundry bag or other designated area for easy access (NOT thrown under your kitchen sink!).
If you want to fold your rags....um, okaaay, but my thoughts are...they're rags.
Many professionals swear by microfiber. Some cleaning services will sometimes assign different colors for different types of jobs; bathroom, windows, dusting, etc. There are different levels of quality, but a decent set of 24 will cost under $20 and can last (with proper care) for many years to come.
Being a green freak, I have been quite reluctant to convert to microfiber from cotton.
Cotton (the end product) is ultimately biodegradable (depending on what it was used for). Microfiber is essentially a "plastic" cloth and once discarded...will be there for millennia, unless sometime in the future we figure out what the f**k to do with all our plastic crap.
Being a purist can sometimes be short-sighted. While cotton is a natural product, growing cotton is very earth UN-friendly and is water and pesticide intensive. Realistically, this comparison ends up with choosing the lesser of two evils.
I have switched to microfiber. Here's why...
Cotton rags need to be replaced often with repeated use and laundering. They can be poorly sewn, unravel, and tear easily which can "catch" on corners or edges causing mishaps.
The downside to microfiber is that (this is the reason they work better than cotton) the micro fiber actually picks up things and keeps them. They're kind of like tiny fishhooks. Say you wipe a countertop where a wine glass shattered. That cloth will pick up any invisible shards that may have been missed. After laundering, if that same cloth is used to dust your furniture, those bits of glass are still there and can scratch! No bueno! This can be a costly mistake for professionals.
Microfiber needs to be washed completely separately to stay effective because it will also pick up fibers from other fabrics (imagine your lint trap on your dryer).
Kind of a PIA, but not an insurmountable challenge.