Genoa doesn't try to charm you the way Florence or Venice do. It's a working port, the busiest in Italy, and it ran its own maritime republic for more than seven centuries. Petrarch called it la Superba, "the superb one", and the nickname stuck because the money was real: the Bank of Saint George, founded in 1407, is considered the oldest state deposit bank in the world. That wealth is still on show in the palaces of Via Garibaldi and the other Strade Nuove, a UNESCO site since 2006, and in one of the largest and most crowded medieval centers in Europe, a maze of stone alleys the Genoese call caruggi.
It's a city of contrasts, best taken slowly. A Baroque palace opens off a scruffy lane, a Renzo Piano aquarium sits beside a fried-fish stall, and the sea shows up at the end of every steep street. Four days cover the city properly, and the fifth goes to the Ligurian coast. Two days would make you choose between the palaces and the sea, and five let you have both.
Day one is the cathedral and the caruggi, day two the Rolli palaces and Strada Nuova, day three a day trip to Camogli and San Fruttuoso, day four the old port and the Lanterna, and day five the Righi hill and Boccadasse. Start on a Tuesday: on Mondays only one of the three Strada Nuova museums is open, and most of the weekday-only sights fall neatly on days one to four. Genoa is also a good way into Italy from the northwest,.
If the Cinque Terre are why you're coming, swap day three for them: an InterCity from Principe reaches Monterosso in about 1 hour 20 minutes, and you'll need a Cinque Terre Card and a booked slot for the Via dell'Amore. The plan chooses Camogli and San Fruttuoso because the same day buys a boat ride, an abbey and a quiet harbor instead of hours on trains, crowds and trails that are often partly closed. The full picture is under If you have more time.