Don't run a hot stone under cold water
Okay so here's the thing. Your pizza's out of the oven, it looks good, maybe it even stretched close to round for once, and now you're standing there with a hot stone and no idea what to do with it. This is the lesson where I tell you the one mistake that'll cost you actual money if you make it. Not dough money. Stone money.
I cracked mine on the counter. Loud crack, like a gunshot almost. Ashlie heard it from the other room and just yelled "what did you do." What I did was pull a 500-plus degree stone out of the oven, set it in the sink, and run cold water over it because I wanted to clean it up fast and get to eating. The stone didn't care about my schedule. Thermal shock split it clean, and it went from "pizza stone" to "two pizza-stone-shaped pieces" in about a second and a half.
Why this happens
Stone and steel don't like fast temperature changes. Going from 500-something degrees to tap-water cold is too big a jump too fast, and the material can't handle it. Steel's a little more forgiving than stone because metal conducts heat differently, but I still wouldn't push it. Cold water plus hot cooking surface is just a bad combination across the board.
Same thing applies if you set a hot stone straight onto a cold countertop or a wet cutting board. I know a guy who cracked his setting it on a granite counter that was cold from the AC. Stone met stone, temperature difference did the rest.
What to actually do
Here's the boring, correct answer: leave it in the oven.
Turn the oven off after your last pizza and just let the stone cool down inside, with the door cracked if you want it faster, closed if you don't care. It'll take an hour, maybe two, depending on how hot you had it and how enthusiastic you were with the preheat (and we already talked about how enthusiastic I want you to be with the preheat). That's fine. You're not in a hurry anymore. The pizza's already eaten.
If you need the oven back for something else, or you just can't stand looking at it, take the stone out with a towel or oven mitts and set it on a wood cutting board, a wire rack, something that isn't cold and isn't wet. Let it sit at room temperature until it's actually cool to the touch, not just cool-ish. Then you can wipe it down.
For cleaning: dry brush, dry cloth, maybe a scraper if you've got burnt cheese stuck on there. No soap, no water, not even once it's cool. Stone's porous. It'll soak up water and soap and then that flavor's in your next pizza whether you want it or not. I just brush mine off and call it done. It's not going to look pretty. It's going to look like a stone that's cooked forty pizzas. That's correct.
A word on steel
If you're using a steel instead of a stone, same rule, different reason. You won't crack a steel with cold water the way you'll crack a stone, but you can warp it over time, and you'll definitely rust it if you're not careful drying it after. Wipe steel down while it's still got some warmth left in it, dry it completely, and if you want to be extra about it, a very thin wipe of oil keeps rust off between uses. I don't always do this. I should.
The burn risk nobody talks about
Real quick, because this matters more than the stone does: that thing is still holding heat for way longer than you'd guess. I've picked up a stone an hour after the oven was off thinking it'd be fine and it was not fine. Use a towel or a mitt longer than feels necessary. Your hands will thank you.
The fry sauce thing
Totally unrelated to stones, but since we're at the end of the pizza and everybody's standing around the kitchen anyway — my brother Blake dared me one time to dip a piece of leftover crust in fry sauce instead of ranch or marinara. I made fun of him for it right up until I tried it. It's good. I won't defend it on some pizza forum where people take this stuff way too seriously, but if you're standing in my kitchen and there's fry sauce in the fridge, yeah, I'm dipping. Try it once before you judge it.
My opinion here, stated as a preference and not a rule: I like my crust plain enough that a good dip actually makes sense with it. If you've drowned your pizza in sauce and cheese, there's nothing left for the crust to bring to the fry sauce party. One more reason to keep things thin.
Before next time
Let your stone or steel cool down the slow, boring way at least once before you get lazy about it, so you know what "actually cool" feels like versus "probably fine" feels like. Trust me on this one. I paid the tuition already so you don't have to.