As for Cuban, head to Calle Ocho and try Versailles; it's humble, noisy and a Miami landmark (be sure to have some after-dinner cafe con leche from the sidewalk window to get the full Little Havana experience).
Versailles is THE classic Miami place for cheap Cuban food, but IMHO there are other hole-in-the-wall Cuban places where the food is much better. One is a market/restaurant called El Palacio De Los Jugos, at 57th and Flagler. It's not a sit-down restaurant; the style is more like a cafeteria with big outdoor tables.
Another is Havana Harry's in Coral Gables. I heard (don't know whether this is true) that it's owned by a married couple--one Cuban, one Anglo--that wanted an authentic Cuban place where both of their families would feel comfortable dining.
David's on South Beach is also good for authentic, hole-in-the-wall food. They have one restaurant on Lincoln Road and another on Washington. At least the Lincoln Road one is open 24 hours a day.
Speaking of Lincoln Road, the Lincoln Road Cafe is around the corner from David's. They serve Cuban food that is good (not great) but has the best people-watching in town.
And a slightly trendier version of classic Latin cuisine is in Coral Gables at Chispa. None of these are landmarks in the way Versailles is, but I think Versailles sometimes coasts on the fact that it's a landmark.
On a separate note, the other night I ate for the first time at a place called Shoji Sushi. It's on Collins Ave. between 1st and 2nd. Innovative food, high-quality fish, and not so expensive that it has to be postponed for a special occasion. They had these lychee cocktails that were just fabulous. The restaurant is next door to Nemo, which is another high-end South Beach favorite.