A very short distance from Salzburg (in Germany) is the Eagle's Nest. This house was built at enormous cost as a gift to Hitler. You can take a bus from Salzburg then make your own way up to the mountain top, but I recommend a tour. I used these guys.

The old town of Salzburg is a typical European place with narrow pedestrian-only streets, cafes etc. The Goldener Hirsch is a fancy hotel with at least two restaurants (and maybe more). I went to one that was very reasonably priced and had really nice meal of various sausages. The Hotel Sacher is another high-end place where you can get a Salzburger Nockerl for dessert. The calories will almost kill you, but it's an (expensive) experience and worth doing once in your life. It takes about half an hour to cook this dessert and it feeds at minimum 3 people.

As others have noted, Mozart was born here and lived a fair portion of his life so much of the town is devoted to Mozart and to classical music in general. Depending on when you go, the Salzburg Festival might have something or other going on. If you're into the performing arts, it's well worth checking out but most of the plays etc are in German.

The Sound of Music tour seems a little cheesy on the surface, but it's surprisingly interesting and a good way to see the area surrounding Salzburg. And I have to admit, it really was neat seeing all those places for real instead of just watching the movie.

I didn't go here myself, but I've heard that the Salt Mines are quite interesting and the area quite picturesque in general.

If you have time, take the train up to Vienna for a weekend.

Have fun!