First, La Tradicion.
La Tradicion is located on Calle 60 north of the Centro area. This was where I first had Panuchos and sopa de Lima, two of my favorite Yucatecan foods. When you come down you need to catch a cab here for dinner. I highly recommend it. La Comida del ayer, hoy. Yesterday's food today. It's delicious.
My second restaurant recommendation is Eladios at the beach in Progresso.
My recommendation is to order your drinks and let them bring you endless finger foods. I took a client there with a couple of Yucatecos and we basically all filled up on every type of Yucatecan finger food imaginable (seafood, sauteed/flavored veggies, ......) and when we were done, after two rounds of drinks, paid a grand total of $20 USD for a party of four. Not Bad. And to top it of....
...You get a tremendous view of the beach right at the base of the pier plus all of the gulf breezes blowing across your face. Just sit back and enjoy.
My final recommendation is for people with kids. If you move here and start getting settled in you need someplace to go where the little ones can play and blow off steam while you sit back and talk and eat. We like Los Trompos at City Center on the northside.
Upstairs has adult dining with a large indoor play area for the kids. Downstairs is where adult only parties sit. The food is typical Mexican food from outside the Yucatan, not Yucateco food. It is very popular however with the locals and gets quite busy at most of it's locations around the city. The food is not expensive, the restaurant is always clean and the service has always been very good, especially if you tip 15% or more.
These are just a few recommendations. Hope this helps.