Go to the mercado in Valladolid, right off the main square, and sample as many dishes as possible. Don’t hesitate to use the spicy black sauce. That is the single best introduction to Yucatan cuisine I know of.
Mérida offers a more urbanized variant, with influences from Cuba (the tortas) and Lebanon (kibi, which is like kibbeh). The town has many bad restaurants, go eat at Punto y Coma, a loncheria inside one of the markets, taxi drivers seem to know where it is. Ask for their specialties, and don’t miss Sopa de Lima.
In Cancún, get yourself to El Centro, away from the tourist hotels. If you are stuck on the strip, Tempo offers ten courses for less than $50, the founder chef is from San Sebastian and I would put the quality at that of a Michelin two-star. Otherwise look for small places selling fish tacos.
El cenote Samula was created by the meteor which did in the dinosaurs, today you can swim there. The open air restaurants to its side were the best meal so far.