Across the valley is the alpine massive of Eiger, Moench, Jungfrau. There is a train that goes up there to the glacier which terminates in the highest train station in Europe, again with great views. It involves going back down into the valley via the Stechelberg gondola (the same you just took up the mountain - jut the other way), and taking a short bus ride to the "Jungfrau-Bahn" (Jungfrau-railway) station in 'Kleine Scheidegg'.
One thing you wouldn't wanna miss is a original Swiss Fondue in one of the restaurants in Muerren - just make sure it doesn't explode your budget
I'd say stay at least 2 nights, maybe 3 in Muerren. There is more info at [https://muerren.swiss/en/ this web site]
[[File:muerren-lucerne.png|400px|right]]
From Muerren your trip leads to Lucerne - still in Switzerland. The train from Lauterbrunnen via Interlaken to Lucerne takes about 2:30 hours and is very scenic. It's best to take a Hotel along the lake, but that's pricey too (Lucerne is expensive in general, quite a few of the worlds richest people live here). I'd recommend the "Hotel Schweizerhof" for its nice views and being close to train station and everything, but it's not cheap. Avoid the hotels right next to the train station.
Take a trip with the old steam boat around the lake, and watch for pictures and placards on those very old boats. Many of the US presidents were guests on those steam ships. It's probably best to ask the hotel staff for any special excursions that may be going on, or if there is a evening dinner cruise - those boats often have specials. Otherwise, just book a regular scheduled cruise, one of the stations to get off the boat is Rigi, find a nice Restaurant for a snack or drink, and wait for the next boat to take you back.
Once you're back, the downtown area of Lucerne has many shops and nice restaurants, it's really nice there. I'd say stay at least 2 nights, with an option for a 3rd.