Well, farinata you definitely need to check out Franz and Ko in Molassana, the best farinata in the city.
For pizza and pinsa (even better) go for 23 Febbraio in via della Libertà. Patanegra in Molassana has a great pizza as well (avoid the same in Boccadasse).
Pizza Sugo is a sort of gourmet, but perhaps a bit expensive for what it is.
Do not miss gelato at Profumo in vico superiore del Ferro. Their panera is the best in the city.

The Cook in Genoa, even if it has Michelin star, does not absoltely worth it. San Giorgio in piazza della Vittoria is much better.