There are so many beauties of Genoa to see and discover and it would take a minimum of a week to "get your feet wet" a bit everywhere, but you certainly can't pass up strolling through the alleys and seeing the Cathedral of San Lorenzo, greeting the dolphins in the Aquarium, enjoying an ice cream on Boccadasse beach and eating focaccia.
You can make your own arrangements or choose one of the many tours and visits, including special ones, offered.

Take some extra time to admire the city from above by moving around on the historic elevators and funicular railways, go as far as Nervi for a picnic in the park and fall in love along the Anita Garibaldi promenade, spend a day between Recco, Camogli, and Portofino to the east, and Pegli to the west arriving there by sea.

There are many places to have an aperitif or eat, by the sea or not, that will leave you with great memories, from typical Genoese trattorias to restaurants with international or ethnic cuisine, wine bars and bars for all tastes, whether you are vegetarian, vegan or have intolerances or preferences. Here I will give you some suggestions by pointing you to the places I know directly, not necessarily the absolute best but know that I really enjoy eating and drinking well.
Online, even in the various travel guides, you can find more complete and detailed lists like the one below.

To test different gatronomic offerings and Genoese street food, you are spoiled for choice.