As you use the drain auger to unblock a drain pipe, both the tool and the drain should be getting cleaner as more water is flushed through, as this process is repeated. This, therefore, becomes the first stage of cleaning.
When the snake is removed from the drain, it should be further rinsed with clean water. It can then be wiped with a rag if necessary.
To improve the life expectancy of your snake and to protect from rusting, wipe it down with an oiled cloth. There parts of a closet auger cannot be replaced.
Drain augers can be stored in the garage, cleaning cupboard or wherever you keep your tools and cleaning supplies.