How to Peel Potatoes
I noticed the term "just washed" in the question.

My suggestion is -- don't wash them before peeling.  Wash them after peeling.

Water is what makes them slippery -- if anything, dirt adds traction as you're peeling.

(and I tend to do one end, then the length of it, slowly rotating, then the other end ... no idea if that's faster or less slippery ... but for carrots, I always do the fatter end first, as I find it'd harder hold a peeled skinny side while peeling the fatter end)