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Transportation help - Lucerne, Switzerland to Fussen, German and then on to Salzburg, Austria

I am in desperate need of some transportation guidance on this routing. We are traveling by public transportation for our trip. Family of 4, two children 8 and 13. I am looking for the fastest way to get us there. Car rental option looks super hard being 3 different countries.
Any advice is much appreciated.
FYI - We will continue by train to Vienna, Austria where we return home.

Arrive Zurich
Lausanne - 3 nights
Interlaken - 4 nights
Fussen - 2 nights
Salzburg - 2 nights
Vienna - 2 nights