I use mostly laminate bibs (such as Bumkins) that wipe clean. If they get really messy, I just put them on the washer to go in the next load. As for wipes, I use cloth diapers and wipes, so I use the wipes for DD's face. I just throw them in the diaper pail and they get washed with the next load of diapers. But, you could use disposable wipes as well and throw them away. If you want to keep using a wash rag, just rinse it out after each use and you can hang it up to dry (by a magnetic clip on your refrigerator, over the high chair, etc.) and you can keep using it until laundry day.
That said, I also have a couple terry cloth bibs. I reuse them until they're really messy. Once they get messy, I just lay them by the washer or in the hamper until laundry day.