En route to Salzburg from MUC Terminal 2

You'll need to get to the Main Train Station for your train to Salzburg. There is also luggage storage at the station.

I'm not sure if there is luggage storage at the airport. There are lockers at the main station, as well as a check room. There are also lockers at the Ostbahnhof. Not sure if you can leave luggage in a locker for more than 24 hrs.

The best way to get to Salzburg is to take the S8 (S-Bahn) from the airport to Ostbahnhof, then a regional train to Salzburg. No need to go to the Hauptbahnhof (lmain station) unless you need to leave your luggage in the check room. The S-Bahn station is between Terminals 1 & 2 at the airport. Purchase a Bayern-Ticket (€25 for one person) at the airport from an automat or for 2€ more from a ticket counter at the airport. The Bayern-Ticket will cover your travel by S-bahn to either station and by regional train to Salzburg.

I'm not sure if there is luggage storage at the airport.

There is one in terminal 1, open 24 hours, and another one in terminal 2, open 5am to midnight. Luggage is stored up to 3 months. Details here.

I need to store luggage for two days. Where is this service?

Salzburg's main station has lockers for 24 hours. So maybe you just take the train there straightaway, via München Ostbahnhof like Lee suggests, and just before the first 24 hours are up, move your bags to a second locker for a second day.

Or you can check into the private luggage storage service in Salzburg at the link below (I have no personal experience with them.)

https://www.nannybag.com/en/luggage-storage/salzburg/salzburg-hbf-train-station

That said - it sounds easier, as long as you do not require access to your stored bags, to leave them at MUC airport.

I'm assuming that the OP has a lot of luggage and doesn't want to carry it all the way to Salzburg, and that he will be away from Munich for over 24 hours, so a locker in Munich is not an option. Ergo, storage at the airport is probably preferred; next best option is probably the check room at the Hauptbahnhof.

There are two S-Bahn (S1 and S8) leaving from the airport station. They leave from opposite sides of the platform about 10 minutes apart. If you are going to the Hbf, when you get on the platform, take the next train to leave, regardless of the side. It will get you to the Hbf sooner.

If you leave your luggage at the airport, take the S8 to Ostbahnhof and catch the Meridian (regional) train there. Both the S1 and the S8 go to Ostbahnhof (the departure board shows S1 going to Ostbahnhof, but that's its final destination, after it goes to the Hbf and through town). The S8 also goes to Ostbahnhof, but that is not it's final destination, and it get there sooner.