We just returned from our trip to northern Italy which included stays in Verona, Brixan (north of Bolanzo) ,and Santa Cristina in the Dolomites. We spent about 4 hours in Balzano/Bozen while connecting between a bus and train.
We spent 2 nights in Verona and loved it. We decided if we went back to northern Italy we would definitely stay in Verona longer and use it as a base for day trips, including Lake Garda.
From Verona you can get a train to Balzano or north to the next town, Brixen. Brixen is a beautiful small tourist town. We enjoyed spending the night there, visiting the cathedral and a museum, and walking around the town. Bozen did not feel like a tourist town - more like a small city where it would be pleasant to live. It also has a cathedral, along with several museums, a classical music festival, a nice park with walking/biking paths and some attractions that are a bit out of the city. i don't know if you would need a car to get there or if there were local buses.
Our hotel rate in Brixen included a card that covered many southern Tyrol museums and attractions, a lift pass on the Plose lift (accessible by bus), and local bus/train fare (including fare out to Val Gardena). Bozen hotels probably offer the same program. There are some lifts near Bozen, so I assume there is hiking not too far out of town.
From both Bozen and Brixen, the #350 bus goes to Val Gardena. In both cities there are bus stops right by the train station. A car is not necessary in the Val Gardena area of the Dolomites. It might be for other valleys - I picked Val Gardena for our stay mainly because we could reach it via public transportation. For some loop hikes, a bus is easier because you might start at a lift in one town and come down a lift in another.
If you like churches/art, stop in Padova/Padua (between Venice and Verona) to see the Scravegni Chapel. Tickets are easy to obtain online, but must be purchased at least a day in advance. The chapel is only a 10 minute walk from the train station.
We were in northern Italy for 3 and 1/2 weeks and relied on public transportation the entire time.
I brought home a number of maps and bus schedules so if you have any questions about details, feel free to ask.