In my experience, local trains are not an issue, can’t comment on buses. High speed trains have also been fine (bike in a box), but storage can be difficult - often hard to find room.
Local trains, as Adam says, are no biggie between Verona and Bolzano. You need to purchase a bike ticket if the bike is assembled so that the rack space can be “yours” (tho I’d take a Voile strap or two in case you need to improvise). I just used Trenitalia a couple weeks ago to go from Bolzano to Milan - it was really simple with my assembled bike, I simply needed to remove my panniers (bikepacking trip) so the bike would fit nicely in the rack.
Trenitalia has a useful guide here. It’s best if you get a train that’s outside of typical commuting hours so you can get a space on the racks. From that guide:
“Train staff may not permit passengers to bring their bikes on board the train in the event that the transportation of your bike is deemed to be detrimental to the train service.”
If your bike is in a box it works well on the high-speed trains but can be a real pain on regional trains as the box is basically a really large suitcase in the eyes of conductors. And there’s limited space for odd-sized luggage. It may be best to store your bike box in Verona if that’s your start and finish point for your trip.
Hey Mark, my family and I traveled to Val Gardena one month ago. We took the high speed train to Bolzano and then used the regional buses. In our case, the buses weren’t reserved seats, even though we were the first people there (we just missed the prior bus) we almost didn’t get a seat. We got pushed to the back of the line and there wasn’t room for our suitcases in the below storage, my sons had to hold their bags on their laps.
On the other hand, we later took a FlixBus with ticketed seats, and I recall there being plenty of storage, even though it was full. That may have been the bus design, hard to say.
So it probably depends on when and where you are going. But if you are trying to get on a shuttle type , no ticket bus, I would run for the storage doors!