Rinse potatoes and place them in a pot (do not cut the potatoes!). Fill the pot with cold water, making sure potatoes are covered.
Add a generous amount of salt to the water. Cook potatoes in salted, simmering water (over medium heat) until fork tender but not falling apart. Drain off the water and cool potatoes completely before cutting.
Cut potatoes half or quarters (depending on the size) and add to a large mixing bowl.
To cut fennel: Place the white fennel on its side on the cutting board and cut off the green shoots close to the white bulb. Cut the fennel bulb in half, lengthwise. Place the bulb half cut side down and slice it across, thinly, like you would an onion. Discard the root. Note: you can use the fennel shoots to flavor the water while cooking potatoes. Just add 2-3 stalks.
Slice red onion thinly, chop scallions, and mince dill weed. Combine vegetables and potatoes in a mixing bowl, season with some salt, and mix with the dressing until evenly coated.
You can choose to serve this potato salad still warm or cold.
If you want to serve it cold, don't add the dressing to the salad but refrigerate both separately. Mix the potato salad with the dressing right before serving. This will help keep the fennel and onions crisp. (Note: cut the onions and fennel only about 2 hours before serving or it will start to get wilt.)