I use one dish cloth a day to wash dishes --> counters --> stove top --> the floor near the stove/sink - corner where food prep is done - (in that order), then it hangs to dry. Once it is dry, it goes into the laundry area and gets washed with whichever load is next. (If I am starting a new load of laundry, I will grab the dish cloth and towels from the kitchen, but the timing is generally the way I first described.)
I use a green scrubbie or stainless steel ball to clean the pots and pans first and then wash in the soapy water with the dish cloth.
We have this nifty magnetic box ($2-3 at Ross/Marshalls/TJ Maxx) that is on the side of the dryer where we keep a handful of clean rags for anyone to use for anything. (We have another small pile in the upstairs linen closet. The main stash is above the dryer in a cabinet.) In most cases, used rags get used once and are hung to dry, then washed like the dish cloths. If a rag is used for a greasy or otherwise flammable mess (very rare), then I toss it in the trash. Super messy and/or gross (but not flammable) messes get cleaned up with rags, which are then rinsed WELL and hung to dry, and then sit on the powder room counter until I wash an appropriate load or they accumulate enough for a load of their own (like today

).