An overview of the surroundings.
The ideal stay in Genoa is of three, preferably four days. The must does are:
1. Portofino and its surroundings, with Camogli, San Fruttuoso and, possibly, Santa Margherita: 1 day visit. You can reach Portofino by car, by train or by ferry from the Expo, two minutes from the apartment. You can also hike on the promontory of Portofino, or stroll through the villages, transfering from one to the next by car, train or ferry. Keep in mind that San Fruttuoso is only accessible on foot or by sea.
2. Cinque Terre: 1 day visit. The best way to reach Cinque Terre is by train, with an Intercity to the first village, Monterosso. To reach all the villages on foot it takes about 5-6 hours. The nicest as well as shortest path (30 min) is between Rio Maggiore (the last village) and Manarola.
3. Aquarium and the historic center: 1 day, i.e. half a day for the aquarium and another half for the historic center. For lovers of architecture and history, however, I suggest a whole day for the old town, visiting Via Garibaldi and its museums.
If you want to stay a few days longer, you can not miss Bocadasse (half a day, or better, one evening), reachable by bus or by foot along the waterfront); Villa Pallavicini in Pegli (half day), accessible by train, bus, car, or ferry from the Expo; a quick ride with the Castelletto lift to see the whole city at your feet! The cimitery of Staglieno, one of the largest monumental cemeteries in Europe a real open-air museum.
Remember, finally, that from Genoa you can reach even the French Riviera and Pisa in just a couple of hours, and in both directions there are wonderful stops, such as for instance Dolceacqua, or Sestri Levante, Lerici and Portovenere.