I married an Irish girl, so any sort of soup proclaiming to involve potatoes has to satisfy a very particular set of requirements. It has to be tasty, it has to stand up to time, and it has to have enough salt but not too much salt. I believe after this, there really isn't a lot else to worry about.
I start my potato soup with bacon.
That's right, just regular ol' bacon. I guess you could get some fancy bacon, with pepper or other kinds of meats in it... But I got this stuff from my local grocer, it's cheap, and fries up pretty nice.
I chop it up into relatively small pieces, perhaps a square half inch for those keeping score.
Then I fry the hell out of it.
While the bacon is giving up the goods, I wash and slice four decent sized leeks. I keep a strainer in large pot of cold water to help rinse any sand or dirt from inside the leeks. Slice, dunk, leave in for a few minutes. It's okay if the rings start to come apart, they're going to do this anyway, and the results will be delicious.
While the leeks are in the tub, I remember the bacon. I like mine done, but not completely burnt. I let the transfer of fat and meat stick to the skillet a little so that there's something left for the other ingredients to hold onto.
When the bacon is to the point where it is almost crispy, but still ever so slightly chewy, I transfer the bacon itself to a large stock pot. Leave the liquid fat in the skillet, we just want the chunks in the pot.
That fat is going to be used to cook the leeks. I want these to start the glorious process of breaking down now, so that we don't have to wait for days while it's bubbling in the pot.
Don't be worried if the leek rings come apart, we want this to happen: it increases the surface area of the leek and it provides more space for the mouth-watering bacon grease to coat in and around everything.
I let mine fry for a good ten minutes, or until they start to get a little golden brown.
Four cloves of garlic, chopped messily. Add to the leeks and continue stirring.
Three large carrots, unpeeled. Fuck peeling carrots. If there's a nasty spot somewhere on them, cut that part out, but really, these things have been in the ground. If you're that worried about germs, stop eating everything. 
I transfer the fried leek/garlic combo to the pot with bacon to chill out for a little while, then continue with the aromatics.
I just drop my carrots in the pan with the leftover bacon fat, let them caramelize a little.
We only buy celery hearts. I used fight the higher cost, but really, with regular celery, the outer stalks are typically woody and dried up anyway. My wife is right, it's better in the long run to just get the hearts. I chop five good celery stalks into small pieces for my soup.
And add them to the carrots. This part of the cooking process seems to take the longest, maybe 20 minutes or more depending on how thick you cut your carrots. In between stirrings and whatnot, I start prepping the potatoes.
That's right, I leave the skins on. The skin is the best part of the potato. It has all the character, vitamins, and fiber, and it also adds an interesting texture to the finished product. Besides, we have 20 minutes to get ten of these chopped up, not really long enough to peel each one.I also like to chop the potatoes into non-regular pieces. Make some big, some small, etc. This adds to the character of the soup. If the soup was to be blended I might worry more about keeping everything uniform for cooking time and whatever, but I'm not, so I don't.
Ah, a glazed pan. What a beautiful thing. The flavor in the crud stuck to that pan is what is going to drive the entire soup. I deglazed this pan with some gin, although anything liquid and thinner than what is on the pan will work. While I didn't get a shot of the deglazing process (it's really quick, and smells awesome), the contents of the pan were dumped into the pot with everything else.
I add herbs, spices, and mushrooms at this point. With this soup I added about a tablespoon of ground basil, a tablespoon of oregano, and two teaspoons of cumin. I ground in a bunch of fresh black pepper (I don't know, maybe 25 grind's worth?), and probably 2 tablespoons of salt. The potatoes will eat that salt for lunch, and we have to help bring out the flavor of the bacon. The mushrooms were an afterthought when I glanced through the fridge when putting the carrots and celery away.
I add just enough water to come below the surface of the veggies (maybe four or five cups?) and crank the heat to med-high.
For the next forty minutes, it's stir, cover, let boil, stir, cover, let boil. We want the veggies to break down, the potatoes to give up their starches, and the flavors to all mingle together.
I don't consider the stirring and covering process complete until the potatoes have a "rounded" appearance, the carrots can be smashed into the side of the pan, and the overall appearance of the water to be murky and muddy. This should smell like a cross between dirt and sugar. 
Once the veggies cease to be recognizable, I add enough whole milk so that the consistency is to my liking (I like it thin), and the color to resemble wall spackle. Turn the heat way way down (hell, turn it off if you like), and add salt and pepper to taste.
Some people add stuff to their soup, like sour cream or chives or whatever... I don't, I just add a spoon. Footnote:
All the parts of the veggies that were NOT used in this soup were thrown into a box and will begin the compost pile of my new residence. We moved in October, and lost a terrific compost bin at the old place. So consider the environment before just tossing this stuff out. The nitrogen will help any soil out, and your tomatoes will be more awesome.
