Salzburg to Bern, Switzerland via rail.

No. All the trains from Austria to Switzerland terminate at Zürich, you will need to change trains for Bern.
The day trains run every 2 hours, Vienna - Salzburg - Innsbruck - Zürich.
The night train (branded "nightjet" and operated by ÖBB, Austrian Railways) depart Salzburg 00:22, arrive Zürich 08:20. See brochure here: https://www.oebb.at/file_source/reiseportal/angebote-und-ermaessigungen/Nightjet%20NEU/OeBB-Nightjet-Folder-2017.pdf
08:20 is not the middle of the night, it is the end of the rush hour, no trouble catching a train to Bern.

Is Bern your destination, or are you changing there for somewhere else?

Thanks so much for the very helpful info, Chris. That is exactly the information we needed. We plan to stay one or two nights in Bern and then hope to find our way to the Glacier Express train.

We plan to stay one or two nights in Bern and then hope to find our way to the Glacier Express train.

The Glacier Express runs East/West through the Alps, Chur to Zermatt. Parallel to the route you will take Austria - Zürich - Bern (which follows the lowlands north of the Alps).
I was going to suggest you take a day train Salzburg - Innsbruck - Zürich, as 00:22 to 08:22 is not a full nights sleep (this train is timed for travellers from Vienna), and it is a scenic route.
But now I have a better suggestion, depends on where you are going after Bern.

• Salzburg - Innsbruck - Zürich day train (1 every 2 hours), but get off at Buchs (first station in Switzerland). Then short connection south the Chur. Overnight in Chur
• Next day, Glacier Express Chur - Visp - Zermatt, overnight Zermatt, explore Matterhorn
• Then move on to Bern
• Option B, skip Zermatt, Change at Visp to go direct to Bern.

It all depends on how this fits in with your plans for after Bern.
You can see this all on the Swiss Rail map here: https://www.sbb.ch/content/dam/sbb/de/pdf/freizeit-ferien/ferien-kurztrips-schweiz/internationale-gaeste/sts-geltungsbereich_en.pdf

Thanks again, Chris. I looked up the NightJet after seeing your first post and found out that there were no NightJets for that route. So this info you just posted is even more helpful! I am debating about whether to see Bern, but I definitely want to do the Glacier Express and was wondering if it ran only one direction. Now I know. I will check out the link. Thanks again.

The Glacier Express runs in both directions.
What I think some first time visitors fail to understand is that the Glacier Express, and the other famous tourists trains (Bernina Express etc.) ARE NOT the only trains on these routes. All rail routes in Switzerland, and all stations, have at least one train per hours (sometimes more). That is all the routes on the rail map I posted earlier.
The Glacier Express is an extra train (extra to the normal hourly trains) for tourists, with panorama windows, food and limited stops.
If you use the normal hourly trains you will see the same mountains, but the train will make more stops and you may need to change trains en route.

You can look up train times in Switzerland, and for your train from Salzburg, on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en/

Thanks! I will check out the website.

Note that if you want to see normal trains on the Glacier Express route on the Swiss railways website, you will usually have to put in Andermatt as a calling point, as otherwise the site will direct you to other, faster routes.

I appreciate all of this information!