Bolzano/Bozen and the surrounding region are well connected by public transport. For local buses, regional trains, cable cars, and other South Tyrol mobility services, the most useful source is Südtirolmobil: https://www.suedtirolmobil.info/en/#/.
Google Maps can also be helpful for a quick orientation, but for local connections, timetables, stops, and regional transport details, the Südtirolmobil pages are usually the better reference. Südtirolmobil also offers a mobile app, which can be useful while travelling in and around Bolzano: https://www.suedtirolmobil.info/de/service-und-kontakt/app-suedtirolmobil.
For longer train journeys to and from Bolzano/Bozen, especially connections within Italy, please use Trenitalia: https://www.trenitalia.com/en.html.
Please note that buses and trains do not run 24 hours a day. Within Bolzano/Bozen there are night buses on some routes, and taxis are also available. One local taxi service is Radio Taxi Bolzano: https://radiotaxibz.it/en/.