We were in Salzburg for 4 days in 2013; headed back in 2 months for another 4 days as part of a larger Salzkammergut trip. It is by far our favorite town so far; walkable and charming with plenty to see in the area.
If you choose to head out of town for a day (and I think this is advisable on a Sunday), Konigssee and the Eagle's Nest near Berchtesgaden make a fantastic day trip. They both take some time to meaningfully enjoy, so plan on a long but wonderful day. At Konigssee, park and walk to the boat/shopping area. You can buy a ticket for a boat ride that will take you to 2 stops: St. Bartholoma and Salet. This place gets busy, so getting there early will ensure you don't have to wait in a long line to embark on the boat ride. The lake is crystal clear, and although most boat tours are in German, you'll still get an appreciation for the scenery. They stop the boat and play a trumpet midway through the lake tour, and the sound echoes off the nearby Alps. It's pretty great. The St. Bartholoma stop is home to a baroque church, a casual place serving steckerlfisch, and a sit-down beer garden. I believe the second stop has some hiking trails that lead to the Obersee (we're doing that portion on this upcoming trip).
If you elect to go to Eagle's Nest, I'd advise going on a tour through the Berchtesgaden tourist office. I think we paid about 50 euro/person. The tour provides bus access to the documentation center (the museum and tour of a portion of Hitler's bunkers) plus a full historical account of the significance of the homes on the hillside leading up to the documentation center (many of them were occupied by Nazi Party leaders and Hitler had a home on the mountain before it was destroyed). The museum itself contains information pretty much all in German, so we would have been out of luck had we elected to do this portion on our own. The tour also included the bus ride and tour of Eagle's Nest and a discussion of its history, with about 30 minutes on our own to poke around. I must say that I'm a big proponent of this tour because, in our view, going it alone up there doesn't provide much historical context.
In Salzburg, be sure to check out the Augustiner beer hall. It's classic Salzburg, and the food is delicious and reasonably priced. Enjoy!!!