Place the cut vegetables in a large colander in the sink or over a bowl. Sprinkle them with 1 teaspoon of the salt and massage for about a minute, then allow them to sit for at least 15 minutes. (Beets will stain other vegetables - so if you'd like to keep the colors separated, salt and let them sit in separate strainers.)
Add the vinegar, water, sugar, the remaining ½ teaspoon salt to a small pot and warm over medium heat. Stir until the sugar is completely dissolved and the liquid is hot.
Rinse the vegetables under cold running water for a minute. Drain and squeeze out all extra water. Place in the container(s)** you plan to store them in along with any of the optional chiles or spices. Pour the hot pickling liquid into the container(s), making sure the liquid fully covers the vegetables.
Marinate the vegetables for at least 1 hour before using, or 2 hours for larger shapes. Store refrigerated in an airtight container for up to 1 month.
*Daikon, other varieties of radish, and turnips smell funky when stored in a container for a while. The taste of the vegetables should be unaffected, but it never hurts to nibble a piece to make sure before including them in a dish.
**A wide-mouth canning jar makes it easier to fish the pickles out of the brine.