Hi,
It really depends on your taste, honestly. Genoa is one of my fav towns in Northern Italy but it can be a little bit off-putting since, like all port towns, it's a little bit rough at its edges. However, just use common sense and enjoy it; it has a very nice historic centre, a wonderful aquarium (Europe's second largest) and a truly intriguing museum on maritime history, the "Galata - Museo del Mare".
Alternatively, two places that I always recommend to folks visiting Northern Italy are Bologna and Ferrara. Bologna is just 1h of fast train away from Milan and both are delightful places, filled with history, good cuisine and charm. Ferrara, moreover, is really a jewel of a city. BLQ airport also has some good connections to the rest of Europe.