Help Transportation from Lucerne, Switzerland to Fussen, Germand then Salzburg, Germany

A quick answer, which starts with a question. Any chance you could change arrival airport to Geneva, it would be faster and cheaper to reach Lausanne from there?

Train travel within Switzerland is best checked at national rail operator https://www.sbb.ch/en - the letters SBB are german initials, as you travel around they will become CFF in french speaking areas (and FFS in italian). The journey planner contains all licenced, scheduled public transport in Switzerland. Everything in Switzerland is expensive, transport is no exception. But there are deals for accompanied kids. https://www.sbb.ch/en/travelcards-and-tickets/railpasses/junior-travelcard-children-co-travelcard.html . If you had a Swiss Travel Pass (full or Flex version) your kids would go free so definately worth considering https://www.myswitzerland.com/en-ch/transport-travel/tickets-for-guests-from-abroad.html .

All the travel in your plan is do-able, the only thought I have is that Luzern is not far from Interlaken and it may be possible to move that stay, do you really want that much time in the city?

As with many questions asked here, you haven't said when! The usually simple trip from Switzerland to Fussen will be harder for the next few weeks due to work on the rail tracks, but otherwise OK. And when you leave Switzerland you get some bargains. From Bavarian border town of Lindau to Fussen you could buy a regional transport day pass, they cost €31 for two people and your own kids under 15 travel free. Same story Fussen-Salzburg. For Salzburg-Vienna there are frequent trains with bargain prices if booked online in advance.

Just to clarify. Do you mean Salzburg, Austria or the small German town between Frankfurt and Cologne?