In British English, a chemist’s is a shop where medicines are sold, and usually soap, toothpaste, and other small health and beauty things as well. It is a shortened form of chemist’s shop. The chemist is the person trained to prepare and sell medicine (what American English calls a pharmacist). That is a different job from the scientist who studies chemistry, though the words look the same — and that is the trap. You say I’m going to the chemist’s the way you say to the baker’s: the shop of that person. In the US people say pharmacy or drugstore. Taiwan’s 藥局, and the 藥妝店 that also sell shampoo and lotion, are the kind of place meant here.

藥局（英式說法，也賣日用品；台灣的藥局、藥妝店就是這類店。美式多說 pharmacy 或 drugstore）