Sounds like a great trip.
I'm biased toward Switzerland, but have visited Salzburg. Sadly, never made it to Venice or Varenna, but have spent quite a few days in Locarno and Lugano. My vist to Salzburg was too brief to appreciate the city. Quite few days in Milano, on business, with minimal tourism, but good food.
I hope your budget will let you eat out well in Italy. The food is usually great.
You are wise to allow a decent amount of time at each location, and to travel before the crowds hit in July.
I'm wondering how to get you to the Jungfrau Region while you are in Switzerland, but I don' t think it is possible. In mid-June it is really too early - just finishing Winter at the higher elevations where the best walks are, and the hotels and restaurants are just starting to open up for the Summer in early and mid-June. Luzern is really nice in June, although there are still Spring rains.
You could squeeze it ( Jungfrau Region) in at the end, but it would be terribly rushed. I would not attempt it, even though it beats Zürich by far as the best place to be a tourist in Switzerland.
8 hrs 23 minutes Salzburg to Wengen at 0756, 0956, 1156, then only a half day in the region the next day, because you want to be back in Zürich the night before you fly back to the states. Most flights out are in mid to late morning.
So. I've attached a few pictures from the Jungfrau region to make you wish you could go there, and a few more from around Luzern. There are good train connections to Milano Centrale from Luzern. You go first to Arth Goldau, a major connection point, then get on the main North-South axis to Milano.Plenty of no-change trains on the route from Luzern.
There are some nice sights near Luzern.
The lake is major attraction, as are nearby mountains. The mountain most resembling the other high alpine areas is Titlis.
This thread describes one of my favorite hikes in the area,which is probably not practical on a short first time visit. But , it has some nice pictures:
A good place to eat in Luzern with a range of good foods and prices is the Rathaus Braurei. Terrific location.
Don't worry about the name - das Rathaus is "the town hall." Think oration and berate. ;-)
map.search.ch/?q=Restaurant+Rathaus+B rauerei,+Luzern,+Unter +der+Egg+2&near=662000 ,190000
I like to go up to Hammetschwand on Bürgenstock, for lunch.
map.search.ch/B%C3%BCr genstock,Hammetschwand ?pos=673816,207488&z=32
If it appeals, I can provide details. See attached images.
More about Zürich later. I spend a total of about 8-10 days in Zürich every year when I travel to Switzerland, so I have some thoughts about the city.