Fit a large bowl with a strainer, leaving plenty of room for liquid whey to collect on the bottom of the bowl.
Line the strainer with 2-3 layers of cheesecloth, clean flour sack towel, or reusable cheesecloth.
Transfer the yogurt to the lined strainer, leaving about ½" of space at the top of the material used to line the strainer.
Cover the strainer with plastic wrap and place in the fridge.
Strain for at least 8 hours, or 10-12 hours for a Greek-style yogurt. Stir once or twice to encourage an even consistency.
Strain for 24-48 hours for a yogurt cheese that can be used as a cream cheese substitute.
Transfer the strained yogurt to a clean container and store in the refrigerator for up to 2 weeks. Discard if there are any signs of spoilage.
Transfer the liquid whey to a glass jar with a lid and use in fermentation recipes, or baking recipes to replace water. If the whey is clear and bright yellow, it will store for a month or more. If it is milky, do not use it for fermentation, but only for baking or making whey cheese. Milky whey lasts up to 1 week in the fridge.
Notes
Using a yogurt strainer: If you have a yogurt strainer, you can use that to strain the yogurt and reserve the whey. If you want a clear whey, line the strainer with a layer of cheesecloth OR a flour sack towel. Otherwise, it will likely be very milky.