Hello - I'm taking the train cable car from Luzern to Murren. When purchasing tickets in Luzern, I'm going to specify the BLM route. Am I at that point purchasing fares for all of the modes of travel from Luzern to Murren, or do I have to pay the fare at each stop along the way (i.e. Interlaken Ost, Lauterbrunnen, and Grutschalp)? I haven't decided yet if I'm going to buy the swiss half fare card or the berner oberland regional pass.
Dan, The train route from Interlaken to Mürren is a train, not a cable car, except for a short stretch at the end between Lauterbrunnen and Mürren.
Tickets are all integrated. Even though it is three different railway companies, just buy a ticket at Luzern station to Mürren. This will cost CHF 50.60 full price, or CHF 25.30 if you have a half-price card; or if you have the Berner Oberland card then you don't need to buy a ticket. Berner Oberland Pass validity map here: http://www.regiopass-berneroberland.ch/assets/PDF/BLS-RAIL-Regio-BE-XL-16WEB.pdf
See Jane's question and the answers at https://community.ricksteves.com/travel-forum/switzerland/train-connections-switzerland
just a couple of questions below yours for the reason to travel between Luzern and Interlaken Ost, just before the ride up the hill, via the Brunig Pass route (via Meiringen).
Buy your ticket in Lucerne and ride all the way to Murren without stopping to buy another ticket. Although if you decide to stop off along the way you can do that...then just catch the next train and continue on your way to Murren.