4-5 nights in Vienna is a little heavy.
I did Switzerland in one trip and Bavaria-Salzburg-Vienna in another, in about your time span, but without the travel from Switzerland to Salzburg. I also had a car for Bavaria-Salzburg-Vienna which was a major convenience (we included Hallstatt and Mauthausen Memorial).
If I were planning the trip itinerary, at the end of the Swiss portion, I would train from Zurich to Konstanz, Germany (about 1.25 hours) to then rent a car to drive from there to Freilassing (just across border from Salzburg so the car stays in Germany) and find a stop or two along the way (Neuschwanstein, Garmisch, Munich?) for the Zurich to Salzburg portion.
I enjoy working out itineraries, so here's my $.02:
May 30 - Arrive Zurich; train to Luzern (45 minutes)
May 31 - Luzern (visit to the top of Pilatus is must-see and good alpine slides overlooking the lake)
June 1 - Luzern to Interlaken (2 hours)-Bernese Oberland - Murren and Lauterbrunnen Valley; the Trummelbachfalle (waterfalls in the mountain) are well worth a visit
June 2 - Bernese Oberland - Jungfrau and Grindelwald
June 3 - Interlaken to Bern (half day) to Zurich (2 hours)
June 4 - Enjoy morning in Zurich; Zurich to Konstanz (1.25 hours); rent car and drive to Schwangau (2.5 hours)[dinner in Fussen]
June 5- Schwangau (Neuschwanstein Castle and some hiking; Alpine slides if you didn't do them in Luzern - or really liked it!)
June 6 - Schwangau to Salzburg, visiting Oberammergau and Garmisch-Partenkirchen along the way to Freilassing (3.5 hours); drop the car off in Freilassing then 10 minute train ride into Salzburg (evening in Salzburg).
June 7 - Salzburg - RS walking tour; Hohensalzburg; evening Mozart concert somewhere
June 8 - Salzburg to Vienna (2.5 hours) - morning to lunch in Salzburg - waterfront bike ride to Hellbrunn Palace (rent bikes at Mozartplatz or cheap city bikes near bridge where RS walking tour starts); Evening - Vienna Opera and Kartner Strasse
June 9 - Vienna - RS Tram Tour and City Walk; Hofburg; evening Opera and Kartner Strasse (if you didn't get to the evening before)
June 10 - Vienna - Museums
June 11 - Vienna
This gets you a nice taste of Switzerland, Bavaria, and Austria and a decent mix of:
- breath-taking mountains and vistas;
- outdoors and adventure
- city, music, culture, museums, castles, and palaces
- and good food!
And, for the most part, the itinerary sticks to RS' 2-night stay rule (all except for the 1 night in Zurich).