Cardiff Market has occupied its Victorian building since 1891 and still operates as a proper working market rather than a tourist attraction. The ground floor is food, fresh fish, Welsh cheese, laverbread (the iron-rich seaweed paste that is Wales's most distinctively Welsh food), cockles, and a bakery counter with Welsh cakes cooked on a griddle in front of you.
Upstairs, vintage clothing and second-hand goods. The whole building is free to browse. Monday to Saturday; Saturday is the best day. If you haven't had laverbread with bacon and a cockle breakfast, this is the place to start.