If you arrive by air, you'll be arriving into Salzburg (W.A. Mozart) Airport (SZG) - it's Austria's largest regional airport. If you're flying directly here internationally, you'll probably stop in one of the major European hubs before continuing on into Salzburg. Once you're here - you can grab Bus No. 2 from the airport directly to the main railway station in town. It takes about 20 minutes or so to get there and the fare cost is 2.80 euros for adults. If you're planning to hit the ground running, pick up the Salzburg Card here (see public transportation for more info) and start your sightseeing immediately. There is also a taxi rank in front of the airport and there are dozens of private transfer companies like salzburgtransfer.com and many others.

You could also arrive by train from Munich or Munich Airport. Austria has great rail service and Salzburg's only about 10 miles from the Germany border and about two hours from Munich. The trains are pretty efficient and a number of trains come from countries all over Europe (France, Italy, Germany, etc., ). The main rail station, Salzburg Hauptbahnhof, is located in central Salzburg, about 20 minutes walking from Old Town. The cheapest train ticket to and from Munich is the Bayern Ticket

You could also arrive by train from  Linz or Linz Airport (LNZ). The cheapest train ticket to and from Linz is the Einfach-Raus-Ticket (ERT) 

You can also drive, but once you're in town, it's probably best to ditch the car and use public transportation or just get out and walk everywhere.