For most of its history Salzburg wasn't part of Austria at all. For about 500 years, until 1803, it was a state of its own, ruled by prince-archbishops who were churchmen in name and Renaissance princes in practice, and who paid for their palaces, churches and fortress with salt, the "white gold" mined in the mountains to the south. They wanted a Rome north of the Alps, and they nearly got one. The city only joined Austria in 1816, after Napoleon had finished redrawing the map.
By then its most famous son was long gone. Mozart was born here in 1756 and couldn't wait to leave, and Salzburg has spent the two centuries since making up for it, with his houses, his concerts, a festival and more chocolate balls with his face on them than anyone could eat. Then came Hollywood: The Sound of Music made Salzburg famous in America and Asia, though most Austrians have never watched it. The old town is small, a UNESCO site packed between the river and two rocky hills, and you'll have seen it in two days. That's why this plan uses the rest of the week for the lakes and mountains that are Salzburg's real backyard.
Day one is the old town and the fortress, day two the right bank, the hills and a beer hall, day three a trip to Hallstatt, day four Hellbrunn, the Sound of Music and the Untersberg, and day five the Königssee and Berchtesgaden, just over the border in Germany. For the rest of the country, see Austria.