Lucky you. You know, Genoa proper is a pretty gritty place, like Marseille. I'd be interested to hear back on your experiences there.
The surrounding area is among the most beautiful in Italy. You should see the Cinque Terre and Portofino if you have the chance. I think that the Michelin Red Book is a good source, particularly with regard to the more humble places. I understand that the Slow Food guide is invaluable, though I didn't have it as a resource on my trip a few years ago.
Lodging at the Splendido is super-steep, but I encourage you to dine there if you can. Top notch Genovese/Ligurian food on a patio high above maybe the most perfect lagoon in Italy, which is saying something. The town below is nice, too.
I have no particular advice about where to eat in the 5 terre. This is a part of Italy where the gas stations have good food.
Last, if you make it far enough west, seek out Joy in Alassio, a spectacular club/restaurant on a cliff with seating on a broad lawn under tents. It's the kind of place hipsters here fail to reproduce. The food, mostly local seafood prepared at the chef's whim based on some gentle questioning from the maitre d', was among the best meals of my life. The web site is cheesy, though, I'll admit. But check out the garden pix.
http://www.uds.it/view_locale.asp?id_locale=6490