1. Use an implement like a metal spatula to take off any food that is stuck to the stone's surface. Of course, only do this once you're sure the pizza stone is cool enough to handle.
2. Never use dish soap on a pizza stone. The pizza stone can be cleaned and rinsed entirely with water. With a clean sponge, wipe away any remaining food or grime using only water. Do not try to remove oils that build up — it is fully unnecessary. Leaving the oils on the stone actually helps season the stone, turning it into a slicker, more easy-to-use item.
3. Don't let the pizza stone soak in water for too long. A simple once-over is probably more than enough. If you pizza stone happens to absorb too much moisture when it's soaking in the sink, it can crack the next time you heat it in the oven.
4. Don't worry about the pizza stone getting stained. Stains on your pizza stone are normal and almost unavoidable. Moreover, they're like a badge of honor, or experience points — something that you point to as a vindication of your cooking skills.
5. Place your stone back in the oven when cleaned, or store it in a safe place with very little traffic. You can house your pizza stone in the oven even as you bake other items. Just bake them on top of the stone itself. For heavier items, like a roast, move the pizza stone to the bottom rack before cooking.