Specifically, I'm thinking about someone going into a drug store to get a Coca Cola or any other soft drink. Would it be cold? How would refrigerating work? I can guess that soda fountains would be cold just by drawing from underground water pipes, but what about buying glass bottles?
modern type refrigerators were prototyped in 1911 and became commercially available by 1913. Despite this most homes and businesses continued to use ice boxes which used ice cut from lakes and glaciers to cool. In 1928 Freon was created by DuPont which greatly accelerated the rate of adoption of modern refrigerators through the 1930's and 1940's.
The story of the Refrigerator PDF by the Association of Home Appliance Manufacturers.
You would have an ice man go through town and just drop a block of ice in your cooler every day. It would have a spout to drain the melt water. The average person would use this to cool absolute essentials like milk or anything else that would spoil fast. If you were really wealthy you could afford to keep cold drinks or have ice in your drinks, but that would be a big luxury.
Soda fountains and the temperature of a coke would vary depending on if it's an urban or rural area. Since Coca-Cola was a syrup that they added to carbonated water for many years, I'm guessing they used ice in your drink. It was bottled by 1899 or so the company website tells me.