We have been to the Dolomites 5 times, for a week or 2 each time. Three times without a car, one time with, and once with a small group for guided hiking. It’s all good, but depending on your travel style and goals, one transport mode might be better than others.
We go to hike and stay IN the mountain villages (or up on Alpe di Siusi), not in Bolzano or other valley. We like to stay where we can pretty much hike from our door, or walk to a lift that takes us to the trailhead. For a hiking-focused visit of a week or less, I suggest going straight to the Val Gardena and staying in one of the villages there (we like Santa Cristina but it is too small for some). No car needed; just take the local bus from Bolzano (bus station right outside the train station).
For this type of visit, I highly recommend you get a copy of this guidebook, which will give you complete information on transport, lifts, hotels, trailheads, suggested hikes, etc. The authors make it easy to plan a good visit with minimum hassle or complicated logistics.
https://www.abebooks.com/products/isbn/9798374791570?ref_=pd_detail-1-d_
(I did not post the Amazon link because it is too long, but you can buy the book there).
If, on the other hand, you are thinking of a visit fueled by Instagram photos of the Tre Cime, Lago de Brailes, and other over-hyped sites (my opinion is showing), you will need a car and a good bit of time planning where to stay and how to reach these places. For example, the parking lot at the Tre Cime is at the end of a toll road. The parking lot fills up early, and once full no more cars are admitted until another car leaves. Or you can take a bus from Cortina or Dobbiaco, but the buses fill up too. . . .
We have reached the Dolomites in various ways, by flying into Munich, Milan, Bologna, Venice. Then take the train to Bolzano, where we rented a car the one trip we did that, or caught the Sudtirol bus, or, when there were 8 of us, booked a private shuttle. You can pick whichever airport offers the best price or fits best with your plans. If you are starting in Verona, then probably Milan or Bologna would be my pick. (I am not a fan of the Venice airport).