... without cutting yourself to ribbons.
In case you don't know what an acorn squash is, imagine a gourd-like, squash like vegetable that is about twice as large as both of your hands clasped together in a fist. The skin of the acorn squash is quite thick and waxy, and the surface topology consists of a series of ridges and valleys running longitudinally from the stem. If you look at it lengthwise, it resembles a multi-pointed star. So you cannot just take your peeler and start at it.
Assuming you have your squash on the cutting board, you now need an extra sharp and stout kitchen knife. Like I said, the peel is quite thick, and the flesh is rather hard so a flimsy blade will not work. Cut off the stem and the end opposite the stem (whatever that's called), and then cut the squash in half lengthwise, parallel to the ridges. Cut it so the blade is in the trough of one of those valleys.
Now comes the tricky part. Extra caution will keep you out of the emergency room. Slice the halves lengthwise along the valleys of the ridges so that you end up with several (eight or ten or so) thin wedges. Now grab your potato peeler.
Again, this instrument needs to be very sharp to pare off the skin. I have not tried a paring knife. It may work better, but I doubt it. Take each wedge in one hand, and carefully peel away from your body. Peel half the wedge, turn it around, and peel the other half. It will be slippery, so peel it over something you will not mind it falling on.
Finally, get your kitchen knife and dice away. Now you have a lovely pile of yummy acorn squash, perfect for an autumn soup. If you make a pot, invite me over!