The Salzkammergut is one of the most beautiful holiday regions in Austria, with its name coming from its history. As the premier spot for Salt ("Salz") Mining. As salt was and is important it was under direct control by the Emperors in Vienna - that's where the name "Kammer" came from.
The regional train has a comprehensive service running through the heart of the Salzkammergut and on into Styria. This provides easy access to the region from both Salzburg and Vienna, passing through Gmunden, Bad Ischl, Bad Goisern, Hallstatt and Obertraun.
I would however suggest car hire as the best way in which to explore this fascinating region. This gives you the freedom to explore area such as Gosausee and St Wolfgangsee, not accessible by train.