Where do authentic locals eat?
Well, you might be surprised at how many head for Burger King, Dominos, or Hooters. I suspect that may not be what you're asking, though.
Keep in mind that breakfast is usually a sandwich or a couple of tacos.
One place for morning tacos is the El Sitio on Calle 2 not far from the taxi syndicate offices. The shrimp taco places (El Camarón Dorado, El Chino Marinero II, etc.) do a lot of breakfast business and close by early afternoon.
You can get empanadas de cazón, panuchos, and salbutes at El Resbalón, which is only open in the morning. It's sort of behind Empanadas Los 3 Patitos, which is not as good a place to get empanadas.
We hit up Tamarindos every Saturday morning for their lechón al horno, which is amazing on tacos. Other days they do great cochinita pibil and their relleno negro is good.
The big meal is in the mid to late afternoon. If locals eat out, they may head to Las Palmas (on 25 Av.), La Conchita del Caribe, St. Chuck's Fish Shack (Pescadería San Carlos), or one of many cocinas económicas.
La Choza is usually packed with locals in the late afternoon, after tourist lunch and before tourist supper, for their amazing lunch specials.
El Súper Hit, of which there are 2 along 30 Avenida, has lots of traditional yucatecan dishes such as queso relleno, relleno negro, sopa de lima, etc.
Be aware of dishes associated with days of the week, such as frijol con puerco on Mondays or mondongo on Sundays.
El Muellecito may seem touristy, but it's a great place for tacos and tostadas, especially tacos that are locally hard to find, such as gobernador, del tio, cochitaco, or torito.
Local food is not usually particularly spicy, apart from the xni-pek.