To get around Genoa, you can take a bus, subway, or cable car. These forms of transportation are affordable, and they have fixed prices. The average rate is about €1.50 per one trip, and the travel time is typically 10-20 minutes depending on the route.
If you have an additional day or two, you can consider visiting the Musei di Strada Nuova, the Galata Museo del Mare, or MuMa- Museo delle Manifatture. More time would enable you to explore the old town even further and enjoy the local cuisine in one of the traditional trattorias.
If you are short on time or want to stick to a budget, Palazzo Reale could be skipped or substituted for another museum or gallery.