Many recipes will call for peeled carrots, radish, and potatoes. The skin has many nutrients. My preference is skin on for carrots and radish, and most times on for potatoes, except my potato salad, I prefer those peeled, but for hash browns, scalloped potatoes, fried potatoes, skin on always.
Unless one is a skilled surgeon, you will remove part of the vegetable when peeling. So what you paid for say a carrot, you are throwing part of the carrot away when peeling.
To remove the skin on a potato, wash the potatoes first, using a knife, just cut through the skin in the middle of the potato all the way around, then boil the potatoes as you normally do. Drain and allow to cool to handle. Once the potatoes are cool, using both hands, grab each side of the potato with the cut line in the middle of your hands, twist a little bit and the skin will come, there, a peeled potato without removing part of the potato itself.
For nutrients in the water after you boil vegetables, you can refrigerate that and use for any recipe that calls for water.