We have done a couple of trips to the Dolomites & have had a fantastic time. We are totally reliant on public transport and have found the train & bus services to be plentiful & reliable. We are keen walkers and have found plenty of options in the Fasa (more Italian), around Dobbiaco (more German) and Ortisei (more Italian). The lifts/cable cars at all spots have let us enjoy walking at high altitudes with fantastic views. We rented apartments and they are a great option and fantastic value-the best info for them is usually in the village web site. Look for apartments on the sunny side of town & you can get some great views.
We transited through Bolzano but did not stay there. It seems a nice enough place but I would be strongly recommending getting to a village in one of the valleys to really enjoy the Dolomites.