I have some clean rags that I found in a first aid kit, and I would like to use them as a bandage, but I can't figure out how. Clicking Consume on them does nothing.
In order to use bandages, you need to use the Medical screen. I believe the hotkey is C.
First, take the clean rag out of wherever you have it stored. Then switch to the Medical screen, and find where you're bleeding. You'll be able to drop the rag onto the wound like it's got an inventory slot.
I'm not sure if rags go from clean to dirty (my one character who survived long enough to bandage died shortly thereafter), but if they do, you should be able to remove the dirty rag, and replace it with a clean one.