Salzburg and Vienna for Mozart and Beethoven

If I got this correctly you are not after the music, but searching places connected to the composers.

In Vienna there are numerous houses (30+) where Beethoven lived and worked. He had to move very often because other tennants and his landlords were annoyed by his loud music all day long, especially in his late years when he became deaf.

There are monuments, streets and places named after him, the house where he died, his celebrity tomb at Zentralfriedhof, his original tombstone in a park, one of the restaurants where he dined (Pfarrwirt), etc.

For Mozart it is similar, but less places.

This is in German, but useful (Google Translate will help) because it references everything related to Vienna: https://www.geschichtewiki.wien.gv.at/Ludwig_van_Beethoven
https://www.geschichtewiki.wien.gv.at/Wolfgang_Amadeus_Mozart

Additionally you may search for the places where major compositions had been premiered.

Museums:
https://www.wienmuseum.at/en/locations/beethoven-museum.html
https://www.wienmuseum.at/en/locations/beethoven-pasqualatihaus.html
https://www.wienmuseum.at/en/locations/mozartwohnung.html

Currently closed: https://www.wienmuseum.at/en/locations/beethoven-eroicahaus.html

Salzburg and Mozart:

As I'm sure you know, he hated the place. It's fun to go to his statue on Mozartplatz and chuckle as you consider how no city so honors a resident who so despised it.

I found the Geburtshaus much more interesting than the Wohnhaus, partially because I found it amazingly cool to stand in the room where Mozart was born. As I recall, there were more artifacts in the Geburtshaus, too. The Wohnhaus is largely a reconstruction following heavy WWII bomb damage. It does have Mozart's fortepiano and some documents that may be of interest to someone with a more-than-casual interest. Mozart played in the Salzburger Dom as a part of his duties to the Archbishop of Salzburg; admission is free. The Mozart family burial plot (minus, of course, Wolfgang) is located in Sebastiansfriedhof, which is not nearly as popular as Petersfriedhof, where Nannerl is buried.

The Rick Steves walking tour for Salzburg is quite good. The Salzburg Museum in the Neue Residenz often has very interesting exhibitions that detail the Salzburg experience of historical European events. For example, I saw an excellent exhibition on WWI there a few years ago. I'm sad I missed this year's exhibition on Salzburg's experience of WWII.

As a music conductor/teacher, you may also be interested in a tour of the Festival Halls used in the Salzburger Festspiele. There is a daily tour at 2 pm in June -- the tour is done in German and in English. It was one of my favorite activities when I visited during the 2014 Festspiele/Festival.

Generally , I find that trying to find palpable connections between living quarters and composers of the past ( as well as painters and authors , with several notable exceptions ) is somewhat like chasing apparitions , The Mozart Geburtshaus in Salzburg is a notable exception , as Dave has written . Since you have stated that you will be in each city for two nights , that really translates into one day each , so choose your visits carefully . The composers section of the Zentralfriedhof in Vienna is a good choice , and don't confine yourself to seeking out Mozart and Beethoven , brilliant though they are , Many other notables in Vienna ,. Let us know which ones might hold an interest for you and we can respond

Okay, so a very little visited Beethoven location is https://bbj.hu/politics/brunszvik-beethoven-event-center-opens-in-martonvasar_85137

Go in July and this may be going on: https://www.tripadvisor.com/ShowUserReviews-g6309097-d6013656-r263886172-Brunszvik_Kastely_Beethoven_Emlekmuzeuma_Museo_Memorie_Di_Beethoven-Martonvasar.html

In addition to Mozart and Beethoven, Vienna is of course known for the Strauss family and their music. We visited the Museum of the Johann Strauss Dynasty two years ago. The museum´s rooms feature 14 themes which can be explored in detail together with musical examples via the high quality audio stations. The museum is located in the 9th District, very near the Freud Museum. Take the 'D' Strassenbahn to get there.

Depending on your time frame (always too much to see with limited time), I would suggest going to Linz from Salzburg as a day trip. Mozart also lived in Linz, (the Linzer Sinfonie). I got to that building but it's not museum, just a private residence indicating he had lived there. The other big composer associated with Linz is Bruckner but that's another story. There is more on him in Linz.

Thank you for the help everyone. I really appreciate it.