Travel from Zurich to Salzburg Sept 2019

There is no overnight train. There are a couple of direct trains otherwise you have to change trains 2 times
You can search to see the train rates and times and buy at :
https://loco2.com/en-us

I think the SBB.ch site is worth looking at.

https://www.sbb.ch/en

It takes about 6 +/- hours for the train Zurich to Salzburg.

Good luck.

Thank you for your information, really appreciated. If we take an early train from Zur to Salz, is there a very early train like 7 am from Lucerne to Zur since we are staying in Lucerne? It seems we are better off departing from Zur for Salz instead of Luc to Salz. Do I need to buy in advance the Zur to Salz train ticket in advance or can we buy it when we get there like a day before going to Salz?

You can use the same https://www.sbb.ch/en website to look up all scheduled trains for Lucerne to Zurich

If you want a direct train Zurich to Salzburg then you should book in advance. The price goes up on that route the longer you wait. The direct is a RJ or Railjet xpress train that you can book here https://tickets.oebb.at/en/ticket

Thank you so much for the links and your suggestions, really appreciated! What Zurich station are we departing from for Salzburg? That is after we arrive in our train from Lucerne to Zurich. Pardon these questions because I do not know if there are multiple stations in Zurich as a departure city for Salzburg.

While there are other stations in Zurich, they are minor little things. Zurich Hbf, where you will arrive from Luzern, is the same one you depart from going to Salzburg.

What Zurich station are we departing from for Salzburg? That is after we arrive in our train from Lucerne to Zurich.

In general that is what the rail timetable website will tell you.
If you go to the SBB website: https://www.sbb.ch/en/
And look up Luzern to Salzburg, it will give you an answer telling you which trains and where and when to change.
You do not need to "guess" the interchange points and look up the various legs individually. That is a recipe for not getting the optimium routing.

The morning train from Zürich HB to Salzburg leaves at an ungodly 6.40 AM, yet you can still take it from Luzern thanks to a 5.28 AM train from Luzern to Zürich HB. This gets you to Salzburg by midday. But frankly I'd leave later and arrive more refreshed mid-afternoon.