Tirol and South Tirol and some other parts of the surrounding area have be divided, united, and under various parties for a long time. The history is fascinating but complicated, so I won't go into details here. I will just say that today there is considerable transnational cooperation between Tirol, South Tirol, and Trentino a but further south. For example, the three states and provinces publish a multilingual avalanche report.
There are many beautiful castles, fortresses, and other old-school structures in Bolzano and the South Tirol. We spent a short time at Runkelstein Castle pictured below.
We also strolled by the Maretsch Castle.
There are at least two cable cars that head up into the hills from the perimeter of Bolzano. We elected to head up the Fenuvia San Genesio, an ancient 20 passenger tram that takes you up to San Genesio Atesino Janesien (I guess that's the name of the town). We then explored the town and went for a hike to get some views of the Dolomites and other ranges of the South Tirol.
No visit to Bolzano is complete without a stop at the South Tirol Museum of Archaeology, which would be better named the South Tirol Museum of Ötzi since most of it is dedicated to the history and study of the 5000 year old mummified iceman that was found within 100 meters of the Tirol-South Tirol border in 1991. I actually found the museum to be quite interesting and well worth a stop.
We've found that mixing hiking and historical sightseeing make a good modus operandi for day trips from Innsbruck. The high country is still covered in snow, so the best hiking is down low and one can almost always find someplace interesting to explore within walking distance or short bus/tram ride from a train station. Mix in lunch at a hütte or gasthaus and an afternoon gelato stop and you have yourself a pretty good day. We also have a discount card for half-price train tickets, which makes transit far cheaper than car rental.