Whatever you do, 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.
When making hot-process soap, do your cleanup in stages. While the soap is cooking, clean up the tools and equipment that were touched by lye and raw soap. Always wear gloves and goggles when cleaning up after working with lye.
Lemons can be a great help in cleaning up after a round of oily soaping. Cut the lemons in half and keep them close by so you can squeeze the juice on your skin after you've rinsed away a lye splash with water and add lemon juice to your wash water. Lemon essential oil is a great degreaser.
Wipe excess soap off the tools, then put them in the sink with a generous amount of white vinegar. Use about ½ to 1 cup per sinkload. Add detergent and wash well, followed with another rinse of hot water. If the tools and equipment still feel oily, wash again with detergent and hot water.
After your hot-process soap is done, it is neutral, so you may continue your cleanup without so many safety precautions. Still take care. Don't rinse chunks down the drain, and make sure there is no greasy residue.