authentic mexican !
In Cozumel, If you're downtown in San Miguel, try Casa Denis, La Choza, or Especias around the Plaza. But some of the best eating can be had at the little places in El Mercado or at Las Flamitas on Avenida 25 or San Carlos Pescardario on Avenida 50 between Calle 3 and Calle 5. Or head for the East "wild" side and have a seafood platter at Chen Rio, ceviche or gorditas at Playa Bonita, or ceviche at the Rasta Bar.
In Progresso, go to Le Saint Bonnet...best seafood place in Progresso on the beach and great camarones. Also Los Cocos along the malecon has good fresh seafood fried, it's cash only but not expensive. Stay away from the tourist traps of Buddy's and El Flamingo. Head for the park and the Palacio Municipal and go to El Cordobes for some of th emore local flavor. Or try one of the loncherias off Calle 80.
Meant to add, the bars and restaurants at the piers are not authentic, unless you consider Taco Bell and Chicoplte authentic. Of course the Punta langist pir is nearest downtown so it's an easy walk. At the Internaitonal pier there's only a junior Sr. Frogs. At the Puerta maya pier (used by carnival and sometimes Princess, there's a 3 Amigos bar and a branch of Pancho's Back Yard. PB does have a reasonable decent margarita and their rellenos are very good and lots of folks used to eating at ChiChi's love the food there. but grab a taxi and go to one of the other spots.