The Dolomites

Ortisei in the Val Gardena (Gardena Valley). It's the main town in that valley and well served by buses.
Val Gardena is the bomb.
No trains can go up the mountains, but you can take a bus from Bolzano or from Ponte Gardena (on the rail line) to Ortisei.
http://www.valgardena.it/en/val-gardena/information-services/bus-schedule/

There will be many Proselytes of the Rick Steves' cult who will recommend Castelrotto. Although it's a pretty town in a nice area, that obviously Rick is in love with, however that is not in the Dolomites. It's rather lower in altitude too.

From the south, the trains go as far as Bolzano and then you'll need buses for the rest. We have gone to the Dolomites numerous times using nothing but public transportation and it's been great! Just keep in mind that bus service is very limited on weekends so make sure to plan accordingly.
We love the Seiser Alm, the Drei Zinnen, the Marmolata, and the Langkoffel Group. Towns we like are Brixen (tiny but "real"), watching the wood carvers in the Val Gardena, Bruneck, Toblach, and Ortesei. Cortina has great views but is EXTREMELY touristy...every license plate is from Rome and the money-culture is pretty ridiculous. It is fun, though, to sit on the main street in the evening and watch the procession of furs and jewelry as people promenade up and down. My favorites are the Japanese tourists that have decked themselves out head-to-toe in traditional Alpine garb.
Note that all towns in this area have both Italian and German names...I use the German ones since that's the language I use in the area. Tourist offices are extremely helpful in the area. We rarely make reservations when we go...we just walk into a tourist office, tell them we're using public transport, and they call and find us a place to stay in our price range. Easy! No matter where you go in the Dolomites the views are stunning, the food delicious, and the culture relatively unspoiled, especially the higher you go up into the mountains.

There are only a few railway lines and they follow the deepest valleys, so you never get a feel of the higher Dolomites. The main railway goes from Bolzano up to Brenner pass, stopping en route at Bressanone/Brixen. A branch of this line leaves from Fortezza (a few miles north of Bressanone) to Brunico/Bruneck - Dobbiaco/Toblach and San Candido/Innichen. The top of this line is at Dobbiaco, a very nice place to be but still distant from the most dramatic peaks. Another branch leaves from Bolzano to Merano and Malles/Mals - a very nice line but getting distant from the Dolomites area.
If you want train access these are the places. Buses fill the voids but probably the best way to travel the area is by car.

I agree wholeheartedly with Roberto. The Val Gardenia is very special, easy to get to, and has many options for views on both sides. Buy a Val Gardena pass and you can ride the lifts and buses to your heart's content!

OK...(heaving heavy sigh)...the Rome license plate comment was nothing against Rome or anyone in it. It was simply meant to emphasize that Cortina is a town that seems to have more tourists than residents. It was a comment about the feel of the town, not about those who visit it.

Most places on the Dolomites have more tourists than residents, especially in high season.

Many tourists in Cortina are of the super rich type, probably more from Northern Italy, rather than from Rome.

If you go through Bolzano, I hope you'll have time to enjoy the outdoor market (except Sundays; dried porcini mushrooms make a nice, light souvenir) and the Ice Man museum/South Tirol Museum of Archeology.