Italy 2018

What about Bologna? Definitely has great food! You can day trip to Parma, Ferrara, Ravenna, and/or Padua.

Will you want a car, not want a car, or could you go either way? And how long a stay in one place were you contemplating? This will greatly influence people's suggestions.

With no car, Bologna is a great base. Even though I didn't care for the city itself (most love it), you have a plethora of great day trips. The only issue is that many of the trains leave earlier or later than I would have liked - be prepared to get a lot of 8:30 AM trains out of Bologna to avoid missing half a day in, say, Ravenna.

Padova is another good base. You can day-trip easily to Venice, Vicenza, Verona, or Ferrara, and Padova itself has 2 days of very worthwhile sights (including the spectacular Scrovegni Chapel) as well as a very nice vibe.

I was going to suggest Turin, but I'm not sure how easy it is to get from there to other places in Piemonte without a car, and I assume (but don't know for sure) that Turin would be as difficult a place to have a car as other large Italian cities. If you will have a car, look into staying somewhere in Piemonte. The region has great food, and Turin itself is both very different from other Italian cities and un-touristed.

A friend enjoyed several days in Udine, in the region of Friuli-Venezia-Giulia. Again, it's both un-touristed and very different from other parts of the country. I know he did at least one day trip from Udine, but don't remember details. He happened to hit a food festival (in September), so I don't know about food the rest of the time.

Have you considered Lake Como??

For how many nights?