Cut the potatoes up into small bite-size pieces (try to ensure they're around the same size so they cook evenly). I don't peel them, but feel free to if you want. Add them to a large pot (don't salt the water) and ensure that they're covered by 1" of water. Bring them to a boil over medium-high heat, then reduce the heat and let them simmer until they're tender when pierced by a knife (about 8-10 minutes from when they start to boil).
At the same time, cook the eggs in a separate pot so they're hard-boiled. I simply time 10 minutes from the point where they start to boil. I cool them by running cold water over them for a few minutes. Peel the eggs once they've cooled.
While the potatoes and eggs cook, add the dressing ingredients to a small bowl and mix together until smooth. Refrigerate dressing until needed.
Once the potatoes are done, drain them thoroughly and then return them to the pot they were cooked in. Drizzle them with 1 tablespoon of white vinegar and toss gently with 2 spoons. Let them cool for 20 minutes (I gently toss them again a couple of times to speed up the cooling process by getting some air circulating).
While waiting for the potatoes to cool, chop the other salad ingredients (pickles, celery, onions, and eggs).
After the potatoes have cooled for the 20 minutes (they will still be warm but not so hot they wreck the dressing), add them to a large salad bowl along with the other ingredients and the dressing. Toss until coated. Sprinkle some paprika over top if desired. You can eat the potato salad right away if you like it warm or chill for an hour or two first to let the flavors meld more (cover the bowl tightly).