Hi- have only called there for one day on a cruise. It was interesting, and we enjoyed it, but one day was enough, not a place I'd go to for longer. That said, there were attractions at the harbour, aquarium, pirate ship from pirates of the caribbean, a viewing tower, a rainforest dome etc. There was a hop on/ off bus ride, Columbus' house, some attractive squares and architecture. The streets between the harbour and town were VERY narrow and dark with tall buildings- atmospheric in the day time, but possibly a bit spooky at night. As steve said, there are the cinque terre towns just along the coast (west), accessible by train from Genoa. Going east, Portofino and Santa Margarita about 25 miles by train. Hope this helps.
kate