What’s the best place and way to store yogurt to maintain quality and safety?

There's really only one place to store yogurt to preserve its quality: in a sealed container in your refrigerator.  All yogurt brands have a tamper-evident foil seal. Many have a plastic lid above that, which is handy for resealing and storage, especially in containers that are larger than a single serving.   Tupperware® and similar products are acceptable for storing yogurt if the container it was purchased in cannot be used.
 
Stonyfield Farms cautions that yogurt should not be stored at room temperature for longer than four hours, sealed or unsealed.  Bacterial cultures begin to die off after that period, and the yogurt is not recommended for consumption.  Most federal agencies recommend that foods derived from animal products not be held unrefrigerated for more than 2 hours.  
 
Sources:
National Dairy Council Yogurt Factsheet PDF
Supermarket research at Trader Joe's, Treasure Island locations in Chicago. 
Interview with Stonyfield Farms consumer relations representative. (name not recorded)
Susan Brewer, Ph.D., University of Illinois, Department of Food Safety and Human Nutrition