Here are some of the places that I like to eat at driving up and down the Keys. I am most familiar with Key West and the lower Keys but have eaten at various Key Largo places.
In Key Largo the Marlin is a good Cuban restaurant and I have heard good things about the Num-Thai restaurant up there. Sushi Nami is good sushi.
In Islamorada the Islamorada Fish Co. is good with an excellent version of Key lime pie. Manny and Isa's is a Key classic and the BBQ joint is pretty good.
In Marathon Keys Fisheries is excellent and Burdines is good for burgers and such. Barracuda Grill is supposed to be very good but I have not eaten there in several years.
On Big Pine Key there is Rob's Island Grill that serves typical bar food that is good. Pizzaworks has very good pizza. There is a tiny sandwich shop called Aroma's that makes excellent sandwiches and Coco's Kitchen is good cuban/latin food.
On Ramrod Key Boondock's has good bar food and ambience since it is a giant chickee hut. There is also 5 Brothers grocery that makes great cafe con leche and cuban sandwiches.
On Summerland Key there is Fish Cutter's that makes good fresh fish and seafood fare. It is a fish supplier so everything is fresh off the boats.
On Cudjoe Key there is the Square Grouper that has a wide variety of dishes and there is Coco's Cantina which is one of the last Keys classic places to eat. It serves up cuban/latin food and also Thai (actual thai cook for that). It doesn't look like much but definitely worth a stop.
On Saddlebunch Key there is Baby's Coffee that has fresh roasted coffee and a good assortment of chocolates, wine, and gourmet items.
On Stock Island there is Chico's that has a nice take on Mexican food. There is also the Hickory House back in the backroads of the island that is worth finding.
Key West: There are probably more great restaurants in Key West than most cities ten times the size because of the tourism. Here are some to try; Michaels, El Meson de Pepe, Antonia's, Mangia Mangia, Camiles, Goldmans Deli, Joses cantina, PTs, Meteor Smokehouse, Ambrosia (sushi), Oragami (sushi), Hot Tin Roof, Croissants de France, Alices, Louies Backyard, Cafe Marquesas, and Blue Heaven. These are just some of my personal favorites and there are plenty more to choose from!