If you get tired of the city you can visit the ice caves south of Salzburg, or the salt mines' (http://www.salzwelten.at/...) close to the city.
Oberndorf, a small city next to Salzburg where Silent Night, Holy Night, the famous Christmas song, was composed and premiered. You can see the Silent Night Chapel (http://www.stillenacht.info) there.
For people who prefer nature to sights, hiking on the Untersberg (http://www.untersberg.net...), a trip to Salzkammergut (http://www.salzkammergut....) or the Königssee (http://www.koenigssee.com/), or a visit to Bad Ischl (http://www.badischl.com/) may be interesting.
Freilassing is a small German town just across the border, 10-15 minutes by train. Some locals prefer to shop here as goods are slightly cheaper than in Austria.
Salzburg is about 2 hours from Munich and Passau, 4 hours from Vienna, and is only a stone's throw from the Alps.
