Bolzano

I liked Bolzano a lot! It's a large enough town to have museums, including one for Otzi the Iceman, and plenty of restaurant options, etc. with easy walking distances between. You'd probably take a train to Verona and then up north to Bolzano. But why are you going? I stayed there because I wanted to hike nearby but it was near the end of the hiking season and I wanted to be in a town with option if the weather turned. That's probably not an issue in June so if the goal is mostly to see the Dolomites and do some hikes, check out Ortisei or other places recommended on the forum. I think Bolzano is worth the stay but if hiking is the primary reason to go to the area, it might be better to stay in the mountains closer to the trails.

I would think that Orteisi would be preferable, car rental at the Bolzano Airport.

As recommended by KC, go to Ortisei instead. Take the train to Verona where you switch to a train to Bolzano. From there it is about a 1 hour bus ride to Ortisei, or about €90 by taxi. (That’s what we do every trip now, taxi.) once there, your lodging will give you a pass for buses all up and down the Val Gardena including a return to Bolzano. Be aware that the hiking season and all transportation options (summer buses, all of the lift) really don’t start up until almost mid-month. When is your trip, exactly?

If you look at the top of this page you'll see a Search bar. If you do a search for Bolzano you'll see that there are more than 1600 postings. This site is an amazing resource. You might also consider searching for Dolomites to gather more information on the area in general.