Hmm, I like Genoa. I find it's similar to many northern British towns, with a Mediterranean feel. True there are rough areas, but there's fun to be had and places worth visiting. As a 'simple tourist' with little Italian it could get strange and the other posters are correct to point out other, more pretty places, but there's no guarantee you'll be impressed / disappointed anywhere you visit. It's down to your sense of adventure.
Considering the history of the place, you could fill a weekend wandering around the museums and cultural sites, shops, restaurants and importantly the stadium and bars