Nancy,
A few more comments......
To answer your question, Mürren is a car-free village so there are no taxis or buses. I have seen a few small SUV's belonging to local farmers, but those are the only vehicles I've ever seen there.
As there are two routes to reach Mürren, you'll have to specify which route you're taking when you buy the tickets. That trip is a bit pricey, but if you have one of the Swiss passes, that will save some money. This provides further information - https://www.ricksteves.com/travel-tips/transportation/trains/switzerland-rail-passes .
From Lauterbrunnen, you'll board the Post Bus (bright yellow, you can't miss it) for the trip to Stechelberg. When you arrive there you'll see the Schilthornbahn terminal. I believe there's an elevator from street level but I can't remember the details as I used the stairs. When you arrive in Gimmelwald, you'll disembark from one cable car, walk across the platform and board another one for the trip to Mürren.
When you arrive in Mürren it's a fairly short walk to the next cable car station. This video will give you an idea what to expect - https://www.youtube.com/watch?v=Cwp1RBMukeI . While visiting Piz Gloria, you may want to stop for a nice hot meal in the revolving restaurant and watch the stunning view pass by the windows.
I believe that construction on the Schilthornbahn is underway at the moment, but I don't think that will present any problems for your trip.
One other thing to mention is that the Schilthorn is at an elevation of about 10,000 feet, so if either of you suffer from altitude-related problems, you might want to have a chat with your doctor about that. Most people don't have any issues for short duration visits.
