One of the best views of Genova is from the Castello d'Albertis. This castle sits atop a massive hill called Monte Galletto, just behind Genova Piazza Principe, one of the main train stations in town. It offers amazing views of the current port of Genova, the old port-turned tourist destination Porto Antico, and part of the historic center of Genova.
More than just a nice view, though, the castle is a worthy tourist destination in its own right.
The castle was built in the late 19th century (finished in 1893) as a private residence of the prominent navigator and sea captain in the Royal Italian Navy, Enrico Alberto d'Albertis. It was designed in a gothic revival style, inspired by Florence and the Aosta Valley in northwest Italy. It features gorgeous arches, sculptures, and a stunning mixture of brick and light stone. It's quite pretty, architecturally dynamic, and the terraced park out front is perfect for views of the city. The castle is also now home to the Museum of World Cultures (Museo delle Culture del Mondo). D'Albertis donated the castle to the city of Genova upon his death in 1932, and the museum opened in 2004.
One final and super fun part of this visit was actually getting there. The easiest way to get to the Castello d'Albertis is via one of the most unique forms of public transportation I've ever been on. Located just across from the Genova Piazza Principe station, on Via Balbi, is the Monte Galletto funicular. It's one of four funiculars in Genova, but this one is special. (A funicular is a railway on a steep slope, like a gondola lift on rails or an elevator on an angle.)
In a feat of engineering I can't begin to understand, this funicular starts as an extremely normal (albeit half glass) elevator in an ordinary building at the top of a hill. Once the doors close, it travels down about twenty stories through a hole in the earth like you're going into a coal mine. Then, it gently lands on a set of rails, the elevator somehow detaches (???), and it travels horizontally through a long tunnel like an oddly small driverless train. It was awesome. For the extremely affordable price of 2€, you can experience this charming oddity and get to D'Albertis Castle without climbing a massive hill.