Travel route from Italy to Austria

Lugano is not in Italy, it's in Switzerland. The fastest way to Hallstatt is via Liechtenstein and Munich.
You can go through Italy if you want, but it's longer. The route you would take is via Milan, Verona, Bolzano, Brenner pass, Innsbruck, Southern Germany, then to Hallstatt.

www.bahn.com says it is a 12 hour journey with 3 or 4 changes. Lugano to Zurich to Salzburg to Attnang-Pucheim to Hallstatt with a double "t" on the end.
Air would be a bit difficult with some out-of-the-way flights for plane changes. Hallstatt is pretty remote.

You ever hear that saying that "you cannot get there from here." Well, it almost applies. Hallstadt is east of Salzburg.
There are no flights from Milan to Salzburg or anywhere close to Hallstadt.
You could take a train to Milan and then to Conegliano where you'd change trains for Salzburg.

Or, you could go by train Lugano to Zurich where you'd change trains for Salzburg.

The best would be to go by train to Zurich and take Intersky (air carrier) to Salzburg. Rent a car there to get to Hallstadt.

Do you have any other place you'd like to see instead of Hallstadt?

Best connections are ..
Train Lugano to Zurich
Railjet Zurich to Salzburg
Bus connections to Hallstatt
It's too long for one day - overnight in Zurich or Salzburg.
Or to fly Lufthansa from Milan to Salzburg via Frankfurt

We're considering breaking up the Lugano-to-Hallstatt trip into two days, riding the Bernina Express bus to Tirano, then the train to Chur and spending the night in there. If so, what would be the best train route from Chur to Hallstatt?

mb,

Breaking up the trip to Hallstatt to two days would be a good idea, and staying in Chur for one night would be good.

The "best train route from Chur to Hallsatt" will depend to some extent on your definition of "best". There are numerous trains on that route, but you'll have to decide whether you want more changes or a shorter travel time. One of the shortest trips is a departure at 07:12, arriving at 14:47 (time 7H:35M, three changes). A slightly easier trip is a departure at 07:12, arriving 15:24 (time 8H:12M, two changes). You can research all the trips on the BAHN.DE website.

Train Chur to Zurich http://www.sbb.ch/en/home.html
Railjet Zurich to Salzburg
OBB Austrian Rail offers premium train service from Zurich, Switzerland
http://www.oebb.at/en/Services/Trains_and_equipment/Our_trains/OeBB_railjet/
Bus connections to Hallstatt

You can short cut a lot of that by picking up the Railjet in Sargans, which is just a 20 minute (max) direct train from Chur.

So you can, with a 5 minute connection.

Chur dep 11:12
RE 4870 RHEINTAL-EXPRESS
Sargans arr 11:32

Sargans dep 11:37
RJ 165
Salzburg Hbf arr 16:03

Duration: 4:51; runs daily