Most cookbooks I've read (and my home ec teacher way back when) have told me to boil white or red potatoes whole, and cut them to size for potato salad after they cool enough to work with. Hello. This is summer. I don't want to boil things for half an hour or so, and then have them sitting around radiating heat into my kitchen, or be chopping anything that's warm if I don't have to. Besides which, boiling whole potatoes takes finesse and luck, otherwise you wind up with potatoes that are mushy on the outside by the time the inside gets cooked. Besides which, I don't know about you but I can't always tell if a potato has bad spots until I peel it and/or slice it.
I've done a bit a research, and as far as I can tell the reason for boiling potatoes whole, in their skins, is to reduce the loss of nutrients, particularly vitamin C. But the same cookbooks that urge you to boil potatoes uncut and in their skins tend to also urge you to save nutrients and reduce vitamin C loss by boiling vegetables in relatively little water and minimizing cooking time. Why that wouldn't work for potatoes beats me.
So, I take russet potatoes (aka baking potatoes), peel them, dice them to the size I want in the potato salad, and boil them in no more water than is needed, preferably with a bit of butter added. Russets are cheap (around here, anyway), cook faster than the other types of potatoes, and work great in potato salad as long as you don't have to chop them after they're cooked. (They tend to crumble if you try to cut them after they're cooked.)
My home-ec teacher is spinning in her grave, but I'm in and out of the kitchen in a small fraction of the time she'd have me use, and we like russet-based potato salad. For that matter, I cook the same sort of potatoes the same way for mashed potatoes.
For either potato salad or mashed, as soon as the potatoes are cooked I drain them in a colander at the sink, and let them air dry a little before going on to the next step.
I'd share my favorite potato salad recipe, but I don't have one. We eat a lot of potato salad and I make a point of making up a new recipe for each batch so we don't get tired of it. Everything is 'by guess and by gosh', and I just keep adding stuff until it looks and tastes good.
The most common mix is chopped potatoes, chopped onion, chopped hard-cooked egg*, chopped pickles, in a sauce made from Miracle Whip salad dressing and one type of mustard or another, with some salt and pepper. But I've used mayonnaise or plain yogurt instead of salad dressing. I'm forever changing which pickles I use (right now I'm using Bread and Butter pickles). Sometimes I add celery. Or green peppers. Or peas. Or cubed cheese. Sometimes I leave the egg out. Sometimes I put herbs in. I like green onions instead of the other kind, sometimes. Whatever is at hand that seems likely might go in. (I was raised in a household that never, ever deviated from a printed recipe if humanly possible, and rarely tried out new recipes. I'm afraid I have the zeal of the convert when it comes to using recipes merely as starting points.)
*Years ago I happily unlearned how I was taught to make what are called hard-boiled eggs. (Never mind what I was taught - it produced eggs with green-edged yolks.) From a Betty Crocker cookbook way back when, I learned to cover eggs with cold water in a saucepan (have them covered well and good, if possible - like by an inch or more), heat to boiling, remove from heat and let stand, covered, for something over twenty minutes (22 to 24 is what the cookbook said, but I generally round it up to 25). Cool the eggs in cold water (it might take several changes of water to get them really cool). Then tap each egg to crack it, roll between your hands to break and loosen the shell, and then peel under running cold water.