3 day hut to hut hike near Bozen

I liked Bozen/Bolzano a lot. It has a very pretty historic district and some museums of interest, first among them being the one housing the Iceman. However, I can't disagree with the prevailing view here that it's better to stay up in the Dolomites rather than down in the valley. Ortisei is a popular option, but I guess Castelrotto might be more convenient, depending on your hiking route. Those are not the only possibilities.

If you do opt for Bolzano or some other valley city/town, be aware that it can get very hot down there in the summer. A lot of the budget-moderate lodgings (otherwise very nice) do not have air conditioning. You need to get up to a considerable altitude to be sure you won't need a/c in June and especially in July. Or you'll likely need to go up some on the quality/cost scale in order to snag a room where you won't bake in Bolzano. The Parkhotel Laurin, conveniently located near the Bolzano railroad station and the old town, is very nice--above my usual standard, but I was tired of sweltering. The breakfast was German/Austrian-style, with lots of meats and good cheeses.

I can't provide tips on serious hiking in the area, but I can assure you that public transportation is excellent. There's frequent bus service from Bolzano (and Bressanone) up to the mountain villages. When the buses stop, the lifts take over. There's a lot of tourism from the German-speaking lands to the north, and those folks take their hiking seriously. I believe you'll find excellent access to any trail you want to hike.

As of 2015 the tourist offices in the area had a very convenient thick pamphlet with all the local transportation schedules, from trains to buses to lifts. I'm sure the information is online somewhere.

The Gillian Price book Walking in the Dolomites may have a good itinerary for you. Walks 16-22 are in the area just east of Bolzano. Local buses and the lifts/gondolas are great.