I'm almost embarrassed to admit that I'm in my 50’s and have never made potato salad from scratch but that's the way it is. :-) I have two questions for you:
First, cut the potatoes into large chunks (1 to 1 ½ inches square) and they will cook faster in boiling water without breaking down. When you cook the potato whole it often means that the outside will be mushy while the center is not quite done. Start with cold water and toss in the potatoes. It will vary on how long it takes based on your stoves heat (electric and gas can differ too as gas is usually quicker to boil) but usually I check them at 15 minutes. If you stab a piece with the knife and it can be squished or mushed the potato is done. Hard spots mean continue to cook until “fork tender”.
Second, it’s fine to leave the skin on. Many of the potatoes nutrients are in the skin or just under that surface, so not peeling may actually help.