You may want to do this walk in sections, as I did it.
Here is a list of public transport methods of getting to the intermediate stops:-

To Kappl - Bus 4240 from Landeck. To Landeck, Eurostar to Paris, Laria TGV to Zürich, ordinary express to Landeck can be done in 11 hours.

To Gargellen - Train from Bludenz to Schruns & bus to Gargellen. Bludenz is on the main line & has trains from Zürich & Innsbruck.

You don't have to go to Gargellen to start in Austria:-
Get to Klosters (see below). Get off at Klosters Dorf station & take the Madrisa gondelbahn (behind the station). Walk up right of the top station to the contour path to the Schlappinerjoch. This gets you into Austria, just. Then, follow the first walk back to Klosters. You have time to do all this on your first day, if you fly early enough to Zürich.

To Klosters - Train from Zürich to Landquart (trains to Chur) &
narrow gauge train (Rhaetische Bahn) from there to Klosters.

To Davos - As Klosters but stay on the Rhaetische Bahn train longer.

To Arosa - Train from Zürich to Chur. Narrow gauge train from there to Arosa.

To Chur - Train from Zürich to Chur.

To Tamins - Train from Zürich to Chur. Train to Tamins or 2.5 hour walk.

To Bad Ragaz - Train from Zürich to Bad Ragaz (trains to Chur).

To Sargans - Train from Zürich to Sargans (trains to Chur, Buchs or Austria).

To Weisstannen - Bus from Sargans or 3 hour walk.

To Walenstadt - Train from Zürich to Walenstadt (trains to Chur, Buchs or Austria).

To Weesen - Train from Zürich to Ziegelbrücke & bus or short walk (trains to Chur, Buchs or Austria). Not many trains stop at Weesen station but you may be lucky. Check in Zürich.

To Linthal - Train from Zürich to Ziegelbrücke & branch line to Linthal (or Netstal).

To Muotatal - Train from Luzern to Arth-Goldau & bus.

To Schlattberg - Train from Zürich to Ziegelbrücke & branch line to Linthal.
Get off before Linthal at Luchsingen & take the chair-lift.

To Braunwald - Train to Linthal & funicular to Braunwald.

To Elm - Train from Zürich to Ziegelbrücke & branch line to Linthal. Get off at Schwanden (before Linthal) & take the bus.

To Unterschächen - Train from Zürich to Fluelen on Lake Lucerne.
Local train to Altdorf & bus to Unterschächen.

To Burglen - Train from Zürich to Fluelen. Local train to Altdorf & bus (or walk) to Burglen.

To Brüsti - Train from Zürich to Fluelen.
Local train or walk to Attinghausen & cable car to Brüsti.

To Engelberg - Train from Zürich to Luzern. LSE narrow gauge train to Engelberg.

To Wassen - Main railway line from Luzern. Make sure it stops!

To Meiringen - Two ways:- 1. Train from Zürich to Luzern.
Brünig line narrow gauge train (this is still SBB - Swiss Federal Railways) to Meiringen.
2. Train from Geneva or Basel to Bern. Train to Interlaken & Brünig line train to Meiringen.

To Grindelwald - Train to Interlaken (see Meiringen) & narrow gauge BOB train to Grindelwald. Don't get on the Lauterbrunnen one by mistake!

To Wengen - See Lauterbrunnen below & get the even narrower gauge WAB Wengernalpbahn train to Wengen.

To Lauterbrunnen - Train to Interlaken (see Meiringen) & narrow gauge BOB train to Lauterbrunnen. Don't get on the Grindelwald one by mistake!

To Mürren - Train to Lauterbrunnen above & funicular & tram to Mürren.

To Kiental - Train to Bern. Train through Spiez (BLS to Brig) to Frütigen.
Bus to Kiental. There are not many. You may have to hitch & walk a bit.

To Kandersteg - Train to Bern. Train through Spiez (BLS to Brig) to Kandersteg.

To Adelboden - Train to Bern. Train through Spiez (BLS to Brig) to Frütigen.
Bus to Adelboden (too far to walk on your 1st day).

To Lenk - Two ways:- 1. Train to Bern. Train to Spiez (BLS or SBB). Train to Zweisimmen (MOB - Montreux Oberland Bahn). Branch line to Lenk.
2. Train from Geneva to Montreux. MOB train to Zweisimmen. Great views over Lake Geneva. Branch line to Lenk.

To Gsteig - Train from Geneva to Montreux. MOB train to Gstaad.
Bus to Gsteig (or attractive 3 hour walk).

To Col des Mosses - Train from Geneva to Montreux. MOB train to Chateau d'Oex. Bus to Col des Mosses.