If you haven't been to Geneva before, I highly recommend a stop there for a day or two. If you decide to make a stop there, the Hotel Edelweiss is a nice Hotel that's close to the center and reasonably priced.
Anyways, with or without stop, on to Bern. If you get there early in the day, a one night stay may be sufficient (you got more stuff coming up). The Hotel Baren is a nice and reasonable option, that has everything in walking distance including the train station. Make sure you visit the Bear Cage, the Einstein house and the botanic garden (there is a restaurant up there with a nice view over the town, good option for a lunch), and find a nice restaurant downtown for a dinner. Check out [https://www.bern.com/en/things-to-do this link for more stuff to do there].
=== Muerren ===
[[Image:bern-muerren.png|right|400px]]
After your time in Bern, your next stop is in Mürren, Switzerland. Take the train to Interlaken, and change trains to Lauterbrunnen. This trip takes about 2 hours. When you arrive in Lauterbrunnen, walk across the street to the Cable Car station (you'll see it when you arrive there), walk up the stairs and board the next car. The Cable Car will take you to Grütschalp (about 20 minutes). When you arrive at the top, board the small BLM mountain railway which will be waiting. The ride to Mürren will be another 20 minutes or so, and you'll arrive in the "basement" station of the this very small station. Walk up stairs to reach street leveltown without car traffic.
For Mürren it's probably best to make a hotel-reservation a few days in advance. The best place in town is the Eiger-Hotel right next to the train station in Mürren. However, Hotel Bellevue and Hotel Alpenruh are nice too and a bit cheaper. Make sure you get a room with a view- and when making reservations on the web, make double-sure the hotel you're booking is actually in Muerren.
Remember this is a vehicle-free place, however, most Hotels will pick up your luggage from the train station with golf-cart like vehicles. Just get their mobile phone number or email ahead of time, and text them when you get on the Gondola. They will be waiting for your luggage at the Mürren 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. For some more tips, and nice pictures [https://wikitravel.org/en/Lucerne check out this web site].
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=== Insbruck ===
[[File:lucerne-innsbruck.png|400px|right]]
Depending on your time budget, you could go from here to Innsbruck in Austria (and to Munich from there), or go straight to Munich. If you decide to make a stop in Innsbruck, the train takes about 5 hours. A reasonable hotel is the "Goldener Adler" right downtown. If you're lucky, you may get the "Goethe" suite, a room where german playwright J.W. Goethe stayed on a trip before. Just enjoy the medieval downtown including the "Goldenes Daechle" and the hearty Austrian cuisine here. A few more tips [https://www.innsbruck.info/en/innsbruck-city/experience/sightseeing-in-innsbruck.html are on this website]
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[[File:lucerne-munich.png|400px|right]]
If you decided to bypass Innsbruck and go straight to Munich, the train ride from Lucerne via Zurich to Munich takes about 6 hours. An alternative is to take the train to the airport in Zurich, and take a short 1 hour flight from there into Munich. The total travel time is probably about the same.
=== Salzburg ===
Instead of stopping in Innsbruck, you could also take a connecting train to continue from Innsbruck straight to Salzburg. Salzburg is worth a full day at least, you have to see downtown with Mozart's birth place, the park gardens and take the elevator up the hill to the castle. Salzburg has nice Austrian charm, so make sure you stay downtown. I don't have a hotel to recommend in Salzburg, since we always go there just for day trips.
There are frequent train connections to move on from Salzburg to Munich, so don't worry about reservations there. The train from Salzburg to Munich takes about 90 minutes.
As with any of the train connections, make sure you don't take the slow train connections that stop on every village. There are fast (and really fast) trains, but local trains may take quite some time to cover a distance.
Another way to stop in Salzburg is to go to Munich first (see above, direct connection from Lucerne to Munich), and after your visit in Munich take a regular train to Salzburg. You stay in Salzburg for the day, and then board the same night train to Budapest that originates from Munich in Salzburg instead. The night train arrives in Salzburg around 1:20AM, but doesn't leave until 3:30AM. You're still allowed onto the train when the train arrives, this way you can board your cabin and sleep, while the train is being recombined.
=== Next stop ===
[[Southern_Germany,_Munich,_Passau|Munich and the surrounding area ]] is next.