Here are some of my personal favorites.
FOR BREAKFAST :
Try to have breakfast at least once at the Museum. Amazing ocean view and good dishes. Ask for the Spanish menu, as it has more options. My favorite there is huevos venenosos, an omelet with different herbs and chaya, aka the Mexican spinach.
Molinas (inside the Municipal market)
Huaraches and other antojitos offered with many different fillings for a few pesos. Wash them down with a soft drink like many locals do. Or buy fresh squeezed juice from the fruit stands.
Zermatt (Avenida 5, between Calle 2 y 4)
Great bakery with a cute little terrasse. The chocolate croissants are not typically Mexican (but good). For local flavors, try to score a tuna empanada (they sell fast).
If you walk in town in the morning along Avenida 10 between calle 9 and 5, you'll see signs for empanadas. Locals make them at the order. They are usually greasy but oh so tasty. There's also a place that's reknowned for them on Avenida 30, haven't tried it yet.
The Coffee Cozumel (aka the Coffee Bean, Calle 3 between Melgar and Avenida 5)
Great coffee and expresso, huge pieces of cake
and one of the places where you can buy some awesome chocolate made on the island (see below).
FOR LUNCH :
Sabores (Avenida 5 approx between Calle 1 & 5)
Chips + salsa + soup + choice of special of the day served with rice and beans + a big jug of agua de jamaica for 60 pesos, homemade and served in a private home (either inside or in the nice garden).
La Gitana (Avenida 65 around Calle 9)
Tiny cantina which serves awesome free botanas with drinks. Had some delicious bites there, including jugo de caracol (conch broth), a wonderful fish soup and many types of "exotic" (for me) tacos (grilled stomach lining and so forth)
Super tasty and not touristy at all. Sit at the bar if you can to ensure you try everything.
Roberto's (Avenida 65, across the street from La Gitana)
Probably the cheapest lunch menu on the island, starting at 35 pesos. Only about 3 choices per day, served with soup and a drink. Homey unpretentious dishes such as picadillo. The owner, Roberto, is the cousin of Ray from El Moro.
Pescaderia San Carlos (forgot the direccion, back in the hood also)
You choose your fresh fish and how you want it cooked. Also delicious ceviches. Everything super fresh. Huge portions.
El Camaron Dorado (on the transversal road, ideal stop if going to the East side)
Seafood tacos and sandwiches. Their signature dish is the taco de camaron empanizado (fried breaded shrimp). I'm salivating just thinking about those. But they have many other seafood fillings.
El Chino Marinero (downtown, I think on Juarez with Calle 20)
If you can't make it to the Camaron Dorado, they serve good seafood tacos and sandwiches here too.
All the above places are open for lunch only.
FOR DINNER
Los Otates (Avenida 15 between Calle 1 y 3)
My favorite for tacos al pastor and quesadillas. Open from noon till 4 am.
Other taco places often recommanded on the board include Los Seras and El Pique on Avenida 30.
Los 3 Gatitos (on Calle 7 downtown)
Few items on the menu but they're all tasty. Awesome huaraches and salbutes, the typical antojitos of the Yucatan, as well as chicken soup.
Kinta's (Avenida 5 between Calle 2 y 4)
More expensive that the above places and not as typical, but definitely worth a visit. The menu changes seasonally and features an interesting mexican-european fusion cuisine. You have to try the salads. Nice atmosphere (you can eat in the beautiful garden patio), great service.
La Perlita (way back in the hood, forgot exact location)
For seafood. Good and fresh.
Also love Rolandi's for thin crust pizzas and sunsets, but that's not Mexican food.
MISCELLANEOUS
Chocolates Kakao makes great chocolate. Perfect for snacks or gifts. They offer tours of their facility and sell their own stuff. Some of their products are also available at The Coffee Cozumel, Los Cinco Soles souvenir stores and El Maharaja delimarket on Avenida 15.