How best to visit Munich and Salzburg on a 6 day trip flying into Munich Airport in early December

Take the train between Munich and Salzburg - very easy. You can just take local public transportation around each town or walk.

If I go from airport to Salzburg is that better than going into Munich Old Town

No, Salzburg is 2h from Munich by direct train making it a good day trip option. You can walk from the Salzburg train station (turn left when exiting the main entrance to get to the old town (30-minutes)). You won't need to use public transportation here. I suggest sleeping in both Munich and Füssen (2h 15m by direct train).
In Munich I would take a half day trip to Dachau (30-minutes by direct train). I also recommend buying Rick Steves Germany 13th edition guidebook: https://store.ricksteves.com/shop/p/germany-guidebook and you can download an electronic version from his app. A 14th edition will be available in the fall, but don't know if it'll be available before your trip. Do take his self-guided walking tours so you don’t miss anything.
In Füssen I would take a day trip to Neuschwanstein and Hohenschwangen castles (15-minutes). You’ll need a good guidebook to help with the logistics, etc.

If I go from airport to Salzburg is that better than going into Munich Old Town first....

The train trip from MUC to Salzburg would add a 2.3-hr. train journey. If you've just made a lengthy, overnight, transatlantic flight, I suggest you just catch a direct train from MUC into Munich, drop your bags, and go about your sightseeing.

And I would book 2-4 nights in each of those two cities. While a day trip from Munich to Salzburg is POSSIBLE, this round trip subtracts 4 hours of train travel time from your seeing/doing time in Salzburg - and there's enough to see and do there that it's hard to squeeze it all into one day in the first place. Also, you might want to do an outing from Salzburg into the nearby Alps (Berchtesgaden? Werfen and Hohenwerfen Castle or the Eisriesenwelt?)

With 6 nights, I might split them as follows...

Nights 1 & 2 in Munich
(Day 3 sightseeing in Munich; evening train on Day 3 into Salzburg.)

3 nights in Salzburg
(Which provides 2 full days for sightseeing/daytrip)

1 final night in Munich or at MUC airport
(On this last day, you could stop over in Munich if you missed something on your first 3 days there; or you could stop over in between Salzburg and Munich at Herrenchiemsee Palace (King Ludwig structure, on an island in Lake Chiemsee. Store bags in a Prien station locker.)

Can I get back to Munich Airport directly from Salzburg or do I need to go back through Munich itself?

Thank you so much for these ideas!!

Can I get back to Munich Airport directly from Salzburg or do I need to go back through Munich itself?

The shortest routing from Salzburg to MUC airport involves one change of train in Munich at Munich Ost station.

But if you want to stop instead at Munich's main station, you can of course go there instead, even if it takes a few more minutes. Trains from Salzburg > Munich's main station are direct. Then you proceed from the main station to MUC airport on a different direct train.

(I'm not sure what Badger is talking about with "there are no such trains." Some kind of misunderstanding, I think.)

Thank you!

The connection to Munich airport is on the S-Bahn, not on the regular railroad, isn't it? Potentially confusing.

there 2 routes of S-Bahn (trains) between the airport and the centre of Munich. Every few minutes.

There are no direct trains between the airport and Salzburg. You need to change at one of the München (Munich) stations, either Hbf or Ost depending on which train you are on.

the S-Bahn is the regular railway, just a more local train.

Hi. This is a great time of year as the Christmas markets should be up and running. If you have not taken this into account, you may want to research what kind of Advent events are taking place, when and where. For example, on December 5, is the Hallstatt Krampuslauf. I would not call it a once-in-a-lifetime experience because I intend to go again! But it is pretty out of the ordinary. You would have to stay over in Hallstatt. I believe most folks get there from Salzburg. Note, there is a Krampuslauf in Munich, but it is more of a formal parade than an interactive event.

Before allotting time and travel, maybe make a list of what is really calling to you. Dachau does not make my list this time of the year. Castles - will it be cold and icy getting up the hill to the castle in December - maybe. It is nice to be out in the country; but only you can tell if the effort involved is worth it.

I would get a guidebook and augment the info with online searches. It is a fantastic time of year to visit. As others have noted, no matter what, you have to transit through Munich if taking the train to Salzburg or the reverse. I have not done the MUC airport to Salzburg option for awhile and would be tempted, Yes, you will be tired, but you will be just about equally exhausted arriving in Munich at say 3PM as you would be getting into Salzburg at 5. There are arguments to be made for either choice. Your landing time in MUC may be a factor - and then all those things you figure out to do.

Wish I were going.

Thank you. Seeing the Christmas markets and visiting the Cathedral in Salzburg on Sunday morning are 2 of my goals. That is also my challenge! Figuring what will work best. I am older and do not want to navigate hills in the ice or snow so I will probably just stay a designated amount of time in each of the 2 cities. Really looking forward to arriving at each location and then walking carefully.

Hi. I meant to add: factor in how comfortable you are arriving at or navigating somewhere in the dark. I am familiar with Munich and arrived last time around 3. I knew exactly how I wanted to spend my first hours: hotel check-in; ATM location; little supermarket location (with a stop on the way "home" for snacks); visit to Viktualienmarkt for my favorite sausages; a quick stop in Dallmayr; a swing around Marienplatz; and then back to the hotel. Night had fallen while I did all that.

Perfect! Sounds like my kind of plan!!

The S-Bahn shouldn't be confusing. It is the normal train you would use when you leave MUC airport station for central Munich. It will probably look something like you see in this photo taken at Munich airport station:

https://www.bayernkurier.de/wp-content/uploads/2019/04/SBahn-imago-aviation-stock-90380058h.jpg

And when you board the train from Munich Hbf (main station) or Munich Ost station to return to the airport, it will look pretty much the same. At all stations there is an electronic board with departure times, platform #'s and destinations and train types (IC, ICE, RE, etc., and S for S-Bahn.) Use the board to find the next S-Bahn train to "Flughafen München."

Have you thought about renting a car? This would expand the Bavarian region for you. Salzburg is great to visit. There are a number of hotels near the train station in Munich. Easy train trip from the airport to DT Munich. There is alot of in between to see. Especially in 6 days. Depends on what you want to do, too and energy level. We would fly into Munich, stay the night, run around, the next day, drive to Salzburg, hang out 2 days and then drive back to Munich and see the sights in the Black Forest/Bavarian region. Stay near the airport and fly home, trying to get a mid morning to early afternoon flight.

I love Salzburg. It's one of my favorite places. And its Christmas market is great.

I would love to understand your flights a little better. There is no need to give dates. Something like, "I will arrive at the Munich airport on Saturday at 9 am and depart Sunday at 11 am" would be helpful. That helps us understand your timeline better.

I personally kind of hate hotel changes, and I would want to knock out some train travel while jet lagged. So, I personally would head to Salzburg by train on the day of arrival, which I have done twice after arriving on the non-stop Delta flight from Atlanta somewhere between 8 and 9 am. My thoughts might be different if you have an afternoon arrival.

Depending on return flight time, I would return to Munich and stay in the Old Town at a hotel close to a train station that would allow relatively easy access to the airport on the day of departure.

Plan to stay in the cities. Don't want a car. Have been to Germany many times (before Covid) so I have visited many places....just NEVER stayed in Munich or Salzburg because I had a car and staying in cities is a nightmare with a car. Do NOT want to drive during the winter. This is a spontaneous trip with my son.
Arrive in the afternoon on Thursday and leave at noon on Tuesday.
Really am appreciating all the information and ideas.

Staying in the cities sounds most reasonable with your timeframe. Arrive Thursday with overnights in Munich Thursday and Friday. Leave sometime on Saturday for Salz (if the Sunday morning cathedral visit is a definite goal). Saturday and Sunday nights in Salz, returning to Munich on Monday for the overnight in that city/region. One alternative to staying directly in Munich for that last night, that might appeal, is Freising which is a town near the airport. It, too, has a Christmas market and its own history.

BTW, one of my fondest Advent experiences was at Der Alter Pater just off Marienplatz in Munich. On the Saturday night, they held an event called Night Fever, which featured music and a relaxed, candle-lit, experience in the church. Beautiful, peaceful! The church was packed, while outside, just steps away, the riot that is the Marienplatz Christmas market was going on. Such a contrast.

I look up the time for "Old Peter" and see if I can fit it in. It sounds like something I would really enjoy!! Thank you!

So, it sounds like Debbie's schedule makes sense for the time you have.

I have not been to Munich for Christmas markets, but have been to Salzburg during Advent season.

A friend and I enjoyed dinner and a chamber orchestra performance at Hohensalzburg, the castle on the hill above the city.

There is a Christmas Museum that my friend and I enjoyed near the main Christmas market.

The Salzburg Museum often has interesting temporary exhibits.

Will look in to the chamber music and dinner. Sounds really nice! Maybe Sunday night!