Can anyone tell me what is the best way to travel from Munich to Salzburg if I don't want to rent a car?
Frequent, inexpensive trains.
There are hourly (mostly) regional trains run by Meridian that stop at both the Hbf and Ostbahnhof (east station). They take just over an hour and a half. A single person can ride that train from Munich to Salzburg for 25€ with a Bayern-Ticket. Additional travelers, up to 4, are 7€ extra, each person.
The Bayern-Ticket, actually is an all-day pass valid in 2nd class for unlimited travel on regional trains all over the state of Bavaria for an entire day (after 9 am workdays).
There are actually some long distance trains that might make the trip faster, but they probably cost more, and they require train specific, non-refundable tickets purchased in advance. Some don't stop a the east station, which is the best transfer station coming from the airport.