Has anyone toured the Dolomites by public transportation If so where did you go and bus information

I took the train to Bolzano and the bus up to Castlerotto. From there I was able to take buses and cable cars further up. You will need to check details on transport when it is safe to travel as they may have changed.

Trains and buses to almost all places in the Province of Bolzano/Bozen (South Tyrol) run every 30-60 minutes
They are a bit less frequent in the Province of Trento (Trentino)
and rather infrequent in the Province of Belluno.(Cadore, Agordino, Zoldino)

It would be easier to give an answer if you told us where you plan to go and in which season

Here's a current thread on the topic...

Dolomites without a car

In order to make it short: the main bus and local train timetables of the area can be found at:
https://www.suedtirolmobil.info/en/j...es-to-download
https://dolomitibus.it/it/l/orario-dolomiti/index
https://www.cortinaexpress.it/en/
https://www.trentinotrasporti.it/en/...th-us/suburban
linea_29_dal_30.01.2021_al_31.05.2021.pdf
01_CARNIA_-_16092020.pdf
There are of course other, seasonal, bus lines from the airports to the most popular resort towns