To get to each location, your best bet is walking around by foot as most attractions are in the same area. However, you can also consider using the public transport system such as the Genoa Metro or buses which have tickets ranging from €1.50-€2.50. If you want to save time and effort, consider hiring a taxi which can cost up to €15.
If you have more time, consider visiting the Genoa Museum of Modern Art, or taking a scenic cable car ride to the hillside village of Boccadasse. If you have less time, consider skipping the Palazzo Reale and Old Port or combine them into one destination. Remember to wear comfortable shoes and try some delicious street food such as focaccia or farinata.