Honestly one of the quickest ways I've found to cool down is to run my wrists under cold water.
And for my dog, wet t-shirts and wet towels tucked under his groin and armpits. I wish he was a water dog (we've tried wading pools, sprinklers etc) but he's not so other than forcing him to drink enough water, I've found he'll tolerate the wet items on him.
I have a pair of cheap UV protection sunsleeves that I've cut to just cover my wrists-forearms that I wet with a spray bottle when it's crazy hot. Works incredibly well.