Dishcloths: cotton waffle weave, the kind that are in a big pack at the dollar store, all pink because our kitchen is pink. Used to wash one round of dishes (usually two days' worth) and then to wipe the counter until I get out a new cloth for next dishwashing.
Dishtowels: best are the cotton "flour sack" type; they dry dishes well, and they themselves get dry quickly when hung up. Used to dry 1-3 rounds of dishes; when it seems not-quite-clean, it becomes the cleanup towel and we get out a clean dishtowel.
Kitchen/dining cleanup, except for the floor: gently-used dishtowel waiting on the oven handle until needed. Once used for a serious job, it goes into the laundry.
Handtowels: terry. Washed whenever they seem dirty, as well as before and after having guests.
Cleanup of floor spills: Cruddy old towels with calendars from the '80s printed on them that EnviroDaddy's grandma gave him when he got his first apartment.

These go into the wash immediately after use. (For greasy or really gross spills, we use paper towels. If just vegetable grease, we compost them.)
All these get washed with regular laundry. Every couple of months I gather up a bunch of them and soak in oxygen bleach before washing.
Napkins: lightweight cotton. Washed with regular laundry whenever they have noticeable food accumulation or are used to clean up a spill. Before a meal with guests, we put all the napkins from the table into the wash and get out clean ones.
Bathroom cleaning: just like dishcloths only green so they can't be confused with the pink ones.
Diaper wipes: flannel on one side, terry on the other. Moistened with water+tea tree oil kept in sports bottle on changing table.
These get washed with diapers in hot water with vinegar.
Handkerchiefs: single layer of hemmed flannel. In addition to nose-blowing, these are used for all baby wiping that's not in the diaper area. Washed with regular laundry unless we've been sick; then, either washed with diapers or soaked in oxygen bleach before washing with laundry.
Panty liners: flannel. Put into mesh lingerie bag and washed with my clothes.
Pads: various fabrics. Rinsed in cold water, soaked in oxygen bleach if badly stained, then hung on edge of hamper in my closet to dry until wash day. Washed with panty liners.