Whenever I have to peel potatoes, I spend much of the time fighting the slippery, just-washed potatoes and keeping them from shooting out of my hands...what's a good way to peel potatoes more efficiently/easily?
This may sound like a silly answer, but it actually isn't: just do it faster. I've worked in professional kitchens, a lot of the time on prep shifts, and although there are lots of great tricks for prepping vegetables, the single thing that helps the most is to simply concentrate and move your hands faster. Build up your speed slowly so you don't cut yourself. And of course be sure you have a very sharp peeler. But mainly, work on slowly increasing your tempo until your hands are a veritable blur, and it won't seem like such a chore anymore.
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)
I know this might sound crazy, but is your potato peeler old or dull? Sometimes a nice new peeler does the trick. Also, I'll sometimes hold the potato with a paper towel if it's slipping in my hands.
If the potatoes really are shooting out of your hands, you might try something as simple as loosening your grip. When I peel potatoes, I don't even squeeze with the hand in which I'm holding it. Generally, putting a lot of pressure on anything when working with knives or other sharp implements can be risky.
A different idea, don't peel them at all. I can't think of a dish where I wouldn't rather have the earthy, crispy, chewy skin of a potato. Just run them under water and lightly brush the dirt off, cut out any eyes, and be on your way. This is somewhat of a non-answer, but it's how I approach most tedious cooking tasks.
Also, you can hold the potato with a towel, or if you have one, one of those "jar openers" that is just the round silicon grip improver. Those work great!
You might have good luck with holding the potato similar to how a football is held for a field goal kicker - place one end on the table and hold the other end with the fingers of your left hand (if you're right handed), essentially sitting the potato vertically, and peel down from top to bottom. You'll still have to pick it up to get the ends, but you should be able to get 90% of the potato done this way. To keep the potato from slipping off the counter, rest it on a paper towel.
Mary Ann from Gilligan's Island presents and even easier method in this video. Basically, you score the potato all the way around, put it in boiling water for 15 minutes, then drop it into an ice bath...and the skin will just come off. Very cool! Of course, you'll want to reserve this method for potatoes that are going to be mashed or otherwise need to be softened for one reason or another.
If you don't mind buying a device specifically for this there are some pretty cool potato peelers on the market, but it might not be very cost effective unless you eat a lot of potatoes!