One of the many wonderful things about soapmaking is that in many instances, it is a self-cleaning endeavor. After all, it's soap!
Be sure you don't rinse large blobs of gooey soap, either finished or unsaponified, down your drains. They will clog up your drains almost immediately and take a lot to clean out once clogged. Your best bet is to use smart cleanup techniques.
Cold-process soapmaking requires more care with cleanup than the other techniques. Since the soap is caustic all the way through the process and for a few weeks after, always wear goggles and gloves while handling it. Keep track of all the utensils and equipment that you've used with the lye.
After you're finished with a lye-touched tool, place it in the sink. If you add water, be sure not to hit it with a hard stream that will splatter. Keep adding the utensils as you finish, pouring vinegar on them as you go.
After you've scraped the last of the beautiful soap batter into the mold, wipe the inside of the pan and any other tools you've used. You can use paper towels, but you'll go through a lot of them. A better idea is to get towels out of the rag bag and tear them into paper towel — sized soap cleanup towels.
You can use your soap cleanup towels over and over, saving money and resources. Place the towels in a plastic bag for a day or two, then add them to the wash. The soap will have saponified enough for laundry use and will contribute to the cleansing.