In the "tips you won't do" thread, many people are adamant about their paper towels, which made me want to pass on our own tip: thrift store towels. We call them "cr@p towels" and keep a supply in the laundry room.
They're excellent for large and/or messy cleanups, and you can either launder the holy heck out of them, throw them away or compost them, depending on the situation.
We buy white ones so dyes don't get in the compost. Preferably hand towels, but we've also ripped full-sized towels into smaller chunks.