I also use those square sponge-material things like PP linked to for wiping up messes major messes, like spilled food or juice. They absorb a ton and you can rinse them thoroughly.
For drying hands, drying dishes, wiping up water spills, etc., I use these:
https://www.williams-sonoma.com/products/williams-sonoma-striped-towels/ . I have about twenty of them in constant rotation and I've had most of mine for ~20 years. They're indestructable and they wash and dry beautifully in the machine. I don't worry if they get staining-type stuff on them because the stains just come out in the wash.
If you end up with chunks of food on something you want to wash in the machine, just shake off or rinse off the chunky bits first. The remaining soil will wash out just fine.