Do Bologna, you’re a 1 hour fast train ride from Milan and Florence and 2 to Rome. Bologna has fab architecture and amazing history, plus the food is amazing. It’s a city with one of the largest historical centres in the world.
Verona is pretty but small and quite boring.
Milan is actually a gem if you bother doing your research. It does have historical architecture, a medieval and renaissance area (Leonardo lived here, near where he frescoed the Last Supper), plenty of art nouveau, canals (again, designed by Leonardo) with lively nightlife, markets and artists quarters, plus it’s a great cultural hub, in particular for design.
yes it’s a financial centre and also the seat of the Italian fashion industry, but do not detract from the rest. I live here and whilst I love Rome (worked there for many years) I find Milan a more cosmopolitan city.
From Milan it’s a short train ride to lakes Como and Maggiore, plus you’re near the Alps. Turin is another gem and 1.5 hrs away by train, Venice 2.5 hours. On the way to Venice you can stop off at Padova, which is charming, and visit Treviso? Another canal city within ancient walls. In fact you could probably stay in Padova and visit Venice from there, it’s only half an hour away in the train. That might work out cheaper than staying in Venice itself.