Do I drink (喝) or eat (吃) a soup in Chinese?
The idiomatic way to say 'try it' in Chinese is 尝一下，尝尝，尝一尝. 

You can also say 你吃一口(试试/尝尝) and 你喝一口(试试/尝尝) for eating and drinking respectively. 

Your version 你喝喝看 is good too. But 你吃吃看 is not suitable for drinkings and soups. This is different from English.