Colonia Centro Specialty
Churros El Moro
Eje Central Lázaro Cardenas 42, Colonia Centro
This legendary "churreria" has been open in the same locale in the Centro since 1935 when the Spanish owners bought the property and imported the Spanish fried-dough delicacy to Mexico. They've been going ever since, in the same old building, with floor to ceiling tiled interiors, and the same menu of churros and four hot chocolate varieties—the thick Spanish version, traditional French style, cinnamon and vanilla-infused Mexican variety, and Swiss, which comes with whipped cream. There are a couple of new specialties on offer, including their take on the ice-cream sandwich, but nothing beats dipping a plain churro into a mug of hot chocolate. If the Centro is too much of a journey, they have a new stand at the Mercado Roma.
Culinary Backstreets Walking Tour
So this isn't a specialty shop at all: it's actually a website and awesome tour company that organizes several monthly food tours in the Centro Histórico (and will lead you to some of the best specialty shops and purveyors). They've done an amazing job of finding rad street food stands and old-school treasures, and creating a totally enlightening tour that doesn't feel trite or touristy in the least. It's worth booking ahead as their small expeditions fill up quickly.
Dulcería de Celaya
5 de Mayo #39, Colonia Centro
This is one of the most beautiful candy shops in Mexico, if not the world. In the same spot in the Centro for over a century, the space itself is a relic from a time past: The wooden display cabinets, baroque-style mirrors, and even the sign haven't changed in over 100 years. The sweets haven't changed much either, as the Dulceria is one of the only places where you can still find such an array of traditional sweets from cajeta, to mazapan, to cocadas, to an insanely sweet, perfectly moist pay de rompope (eggnog cake).
5 de Febrero #28, Colonia Centro
Conchas, Orejas, Garibaldis, Bolillos, Teleras, etc., El Globo carries all the traditional breads and pastries, and is a weekly and sometimes daily stop in some households. Even though it's a large chain, there's still a nostalgic charm to its shops, the first of which opened in the 19th-century. Staff wear old-school pinstripe uniforms and you select your own breads and take them to the counter on battered pewter trays.