Munich, Vienna, Salzburg

There's plenty in Munich and Salzburg to take up all of your allotted time. Salzburg's a couple of hours east by train, but you should reserve at least 2 days for there.
Vienna is another great large European city worthy of at least 4 days. Skip it this trip.

You need to split your time between Munich and either Vienna or Salzburg. Munich to Vienna is about 3 hours by train one way, while as stated above Munich to Salzburg is about 2 hours one way.

If I were arriving and departing in Munich, I would definitely include Salzburg and focus on the surrounding area - Neuschwanstein castle in Schwangau, Tirol, Oberammergau, Garmish-Partenkirchen, Mittenwald, plus Berchtesgaden and Hallstatt near Salzburg.

While there is plenty to do in just Munich and Salzburg areas to take up your full time allotment, you also specifically mention that you would like to visit Vienna for a couple of days. If you also want to visit Vienna, you have time to make Munich-Vienna-Salzburg work.

Here's a sample Munich-Vienna-Salzburg itinerary that can be done easily by train or by car:

July 19
EVE: arrival in Munich
(sleep in Munich)

July 20
AM, PM, EVE: Rick's Munich walk - Marienplatz, Residenz, Hofbrauhaus, Viktualienmarkt, etc.
https://www.ricksteves.com/europe/germany/munich
(You might be able to squeeze in Dachau this day, if you don't spend much time in museums/galleries. But, I include it later on in the week.)
(sleep in Munich)

July 21
AM: train/drive to Vienna (4 hours)
PM, EVE: Rick's ringstrasse tram tour that encircles Vienna city center, attend an evening opera at the Opera House
https://www.ricksteves.com/europe/austria/vienna
(sleep in Vienna)

July 22 - Vienna
AM: Rick's Vienna walk - St. Stephen's Cathedral, Visit the Hofburg Imperial Apartments.
PM: Kunisthistorisches Museum. Might have time for the Naturhistorisches museum
EVE: Classical concert (or switch July 21 and 22 evenings)
(sleep in Vienna)

July 23 -

AM: Schonbrunn Palace
PM: travel to Salzburg (2.75 hours by train, 3-hour drive) [consider to stop in Melk enroute and visit the Abbey - about 1 hour from Vienna)(a 2-3 hour stop in Hallstatt and lovely drive through the Salzkammergut Lake District are also doable if you are driving and don't mind a later arrival in Salzburg - add about 1.5 hours driving time)
(sleep in Salzburg)

July 24 -
AM: Rick's Salzburg walk and Hohensalzburg Fortress; consider Monchsberg walk as part of this
PM: Mozart birthplace and residence, Salzburg Museum OR consider a river-front bike ride to Hellbrunn Palace (my choice)
EVE: Mozart performance at Mirabell gardens and palace or Marionette theater
(sleep in Salzburg)

July 25 -
AM: Travel to Munich (2 hours by train, 1.5 hour drive)
PM: Dachau concentration camp
(sleep in Munich)

July 26 - Flight departure from Munich

If you drive, rent the car when leaving Munich and try to get a hotel in Salzburg with parking. Though, you can park for 24 hours under the mountain, right next to Salzburg city center for around 10 euros. There's parking near the Opera House in Vienna for about 22-25 euros per day...also about 15 euros/day in the museum quarter.

Note that Mauthausen concentration camp is off the main road between Vienna and Salzburg. Some prefer Mauthausen to Dachau because it still contains original buildings and might be a bit more authoentic. But, it's more difficult to reach. I've only done it by car - a little more difficult to reach by train and bus. https://community.ricksteves.com/travel-forum/to-the-west/mauthausen-vs-dachau

Wow Bradley, your response is amazing and fits exactly what I'm looking for. I'll get onto your suggested itinerary right now.
Thanks again

I agree with David. Skip Vienna this time. I just got back from Munich, Salzburg, and Hallstatt last week Thursday. It's so incredibly easy to take a train between Munich and Salzburg. I spent 3 days in Munich and Salzburg and 4 days in Hallstatt. I wish I would have spent 4 days in Munich and Salzburg and only 2 in Hallstatt. There is so much to do in both Munich and Salzburg. When in Salzburg, go to an information center and purchase a Salzburg Pass. For one fee for the number of days you think you will use the pass, you get unlimited bus rides and free entry into all the museums plus a river cruise that lasts about an hour (2pm only for this, I would advise you get there by 1:40 to secure your spot!).

While in Munich, we took a day tour (through Viator.com) to Dachau, by train. Our guide was awesome and knowledgeable. We left at 9am and were back in Munich by 2:30 - still lots of time to enjoy ourselves!

If you can, when in Salzburg, do a day trip to Hallstatt. It will take one whole day, but it is so worth it. I would go to the salt mine and the overlook - you take the funicular to both. If you like to hike, there are hiking trails. Just note that all food in Hallstatt is cash only and the town shuts down around 6pm!

Your trip is too short for the 3 cities you want to do, at least in my opinion. Have fun - it's a BEAUTIFUL area!

I'm planning a similar trip in May. My daughter will be in Vienna and she will meet us in Salzburg and we will travel back to Vienna w/ her so she can tour us around the city.

Can you check my plan - does it make sense?

I prefer smaller towns to cities. And we want to do some moderate hiking.

Thursday - Arrive Munich
Friday - Explore Munich
Saturday - Travel by train to Salzburg
Sunday - Explore Salzburg
Monday - Day trip to Eagle's Nest (we will rent a car)
Tuesday - Day trip to St. Gilpen? (rental car)
Wednesday - Halstat? Drive the Grossglockner? Werfen? Konigssee? Stay in Salzburg? WHAT SHOULD WE PICK?
Thursday - Salzburg by day then Travel by train to Vienna in afternoon (or should we drive to Melk and then train to Vienna?)
Friday - Enjoy Vienna
Saturday - Enjoy Vienna
Sunday - fly home

The Salzburg Festival will run July 20 through August 30. During the festival, Salzburg becomes the cultural center of Austria with numerous plays, operas, and classical performances. Indeed, Salzburg is the summer home of the Vienna Philharmonic. Tickets are pricey for festival performances, but it's a great atmosphere for one who enjoys such things.

The program will be up Nov 13. Perhaps worth a look before you commit to Vienna?

https://www.salzburgerfestspiele.at/en/#summer2020

IMO I would pick Vienna over Salzburg, but not a day trip. I agree with others you probably should do only 2 cities in total.