Genoa is a lovely city to visit especially when the sun is shining. A good way to see a bit more of the city is to take the train tour leaving from Port Antico.
Alternatively you could wander down through the streets to the port and sit and watch the world go by at one of the many street cafes.
Of course being in Italy you must try the ice cream at one of the Gelaterias. Wander round the port and look at the private yachts some of which come with their own helicopter!
We stayed in the Hotel Continental which was centrally based and in the evening you could enjoy a drink on the rooftop bar which had stunning views across Genoa.
As a first trip to Italy I would recommend a visit to Genoa.