The Traveling Professor runs small group tours to Italy and we do visit Rome, Florence and Venice. However, there are cities that need to be visited to uncover the REAL Italy.
Padua: Only 30 minutes from Venice, Padua is free of the cruise ships and trinket shops that plague Venice. Instead, visit four piazzas (including the largest in Italy) practically free of tourists. Padua boasts the Scrovegni Chapel (much better than the Sistine Chapel, if you ask me), the Basilica of St. Anthony, great food, authentic markets. It is far more affordable than Venice. This is my favorite Italian city and I often visit for 4 or 5 days.
Vicenza: The best food in Italy. Beautiful architecture. Lazy piazzas. A wonderful day trip.
Bassano del Grappa: Only an hour by train from Venice, this small town tucked in the foothills of the Dolomites is differently Italian. Shops are closed for lunch, the slow-going pace is more than enjoyable. Of course you can taste the grappa and even enjoy a German feast. Beautiful countryside.
Verona: OK, it can be a bit touristy. It reminds me of a smaller Rome with a more-intact Colosseum. Fantastic food, good hotels, and plenty of places to shop.
Treviso: Easy to get to by train. We enjoyed luxurious accommodations at a third of the price of those found in Venice. A great place for a lazy Italian weekend with out of this world food and few bothersome tourists.