Genoa or Lugano

For me, the best part of Genoa is the Ligure coast outside of Genoa - which isn't the easiest day trip from Milano. But, if you're interested, you can get to Santa Margherita Ligure in 2 to 2.5 hours by train (via Genoa). It's a quick 15-minute bus ride to Portofino and a nice walk/hike to the Portofino lighthouse. But, that's 4-6 hours round-trip, depending on transport timetables - a lot of travel for a day-trip.

Lugano is closer and probably a better use of your day-trip travel time. Also, snow-covered Alps could provide a fantastic setting. Also, consider Lago di Como or Lago Maggiore on the Italian-side of the border. Visit lake-side villages, like Varenna and Bellagio or the British and German hotspot of Stresa. I've never been in the winter, but Stresa has a cable car that takes you atop Mottarone. I believe there's skiing here in the winter.

Weather and your priorities would be important factors to consider.

Lugano is definitely prettier than Genoa.

Genoa IMO isn't really a tourist town. Not sure what would draw you there - in February I might play it by ear since weather up north could be iffy. But I love snow and cold weather so I think I would head to Stresa or Lake Como - easy to get to from Milan.

If you want a place with more indoor attractions (which you may well, in February), look into Turin. It's got a completely different feel from other Italian cities, and there's the Egyptian collection, the car museum, the film museum, and others.