Travel from St.Moritz to Mürren by train

We are doing this very trip -- at least as far as Lauterbrunnen -- in the fall. We will leave St. Moritz on the 9:15 Glacier Express, arriving Brig at 15:40 where we will change to a train to Lauterbrunnen, dp 15:49/ar 17:25.

To go to Muerren you would have to take the gondola to Grutschalp followed by the train OR bus to Stechelberg and two-stage gondola to Muerren.

Long day, but scenic. I researched my options at www.sbb.ch/en but I was always planning on the Glacier Express so did not look at going via Zurich.

Going via Zurich is faster, 6 1/2 hours with at least 5 changes, more often 6 or 7.
Going with with the Glacier Express almost 9 hours and still 5 changes. The Glacier Express costs a bit more, plus a reservation fee. There are also regular trains on the same route with more changes, that charge about the same as the route via Zurich.

Certainly, more and better scenery on the Glacier Express route, and on the actual GEX a bit more relaxing as it is 6 1/2 hours to the first train change in Brig.

To see it on the SBB trip planner, put in "via Disentis-Muster".

I'm also going to make the trip soon, in the opposite direction, changing at Spiez and then at Brig to the branded Glacier Express train. Any schedule reflecting the Glacier Express is likely to refer to Express, GEX, Panorama car, and/or subject to compulsory reservation. Or just book the reserved section at https://www.glacierexpress.ch/en/ then figure out how to get yourself from Brig to Interlaken and Muerren.

The SBB website ( https://www.sbb.ch/en ) always gives you the quickest route. If you select St Moritz to Lauterbrunnen, it will route you:
St. Moritz - Chur - Zürich HB - Bern - Spiez - Interlaken Ost - Lauterbrunnen - Mürren
6 or 7 changes, takes about 6½ hours. This involves going north, then west round the Alps across the (relatively) flat parts of Switzerland, hence the trains run faster and it is quicker.
You can trace that route on the rail map here: https://www.sbb.ch/content/dam/infrastruktur/trafimage/karten/karte-sts-pass-gueltigkeit.pdf

But, you want to go via the Glacier Express route via Andermatt and Brig (you don't have to take the specific Glacier Express train, you can take any of the hourly trains on that route), You need to add a "via Andermatt" option when looking for trains on the SBB website. And you get the following (plus later trains with more changes):

St. Moritz dep 09:15, GEX 903 Glacier Express, Brig Bahnhofplatz arr 15:40
Brig dep 15:49, IC 8 827, Spiez arr 16:24
Spiez dep 16:33, IC 61 975, Interlaken Ost arr 16:57
Interlaken Ost dep 17:05 , R 179, Lauterbrunnen arr 17:25
Lauterbrunnen BLM dep 17:38, Grütschalp arr 17:42
Grütschalp dep 17:44, Mürren BLM arr 17:58

5 changes, but it takes 8h43, 2 hours longer than via Zürich. You can also trace this route out on the map, and you can see it goes through the Alps. Slow but more scenic.
This is the one you want!

Most of the changes are at the end, from Interlaken onwards. But these are the same both routes, and the trains are all waiting for each other.

About the only places you may have a few minutes wait are at Brig and at Spiez. On both occasions you are changing direction of travel so the connections are not quite so smooth. You can watch all the trucks on the trains roll past at Spiez.