Though I have to admit the main attraction of this remarkably pretty town has to be Mozart and then the historical buildings of the Old Town and the castle.
If you want to know about that aspect of Salzburg (very worth visiting for) then I would advise reading Michael-S's opinion, which describes most of the major sights far better than I would hope to do so. Added to what he says, I would point out that the Franciscan church performs a Mozart Mass at 9am on Sundays (at least it did on the week we were there) which is well worth going to, if only to hear the music as part of a Mass, as it would originally have been played.
There are also numerous concerts going on every night, many with dinner included. We went to one up in the castle, which was a marvellous setting for chamber music. It cost us only AS200 (for students, full price AS500). The Mozart piece they played (Eine kleine nachtmusic) felt rather mechanical but the Verdi and Beethovan which followed were much more expressive and moving. I guess they must have to play Mozart every night, so it's not so surprising they're tired of him.
The castle does cost a lot (~AS50) to enter during the daytime and another 50 to see the museum and interior. The funicular is another AS27, the walk's very steep but shouldn't take much more than 10-29 minutes. I'm not sure it's worth it, though there is a spectular view of the city from the walls.
The shopping in the Old City and Getreidegasse is very good - lots of shops selling just about anything that will appeal to the average tourist. Specialities include: Mozart-chocolate, dolls, pretzals and schapps.
Lastly don't forget the Sound of Music tour. We had great fun going on this (in an ironic way of course ^_^), singing along with the soundtrack as it was played in the coach and acting out scenes from the film when we stopped. The tour takes you to most of the major locations used in the film (those that aren't in Hollywood) - the Wedding Church, the pavillion, the house. You can get the tour cheaper if you're staying in the Youth Hostel, but otherwise it leaves from Mirabelleplatz at 9am and 2pm daily.
Ryanair does direct flights to Salzburg airport, only 1/2 hour by bus from the train station, very cheaply (we got a £4.50 one way flight). The town itself is very walkable - at most 1hr to get anywhere.
We stayed in the "Jugengasthause" youth hostel, which was very reasonable and fine for our purposes. Breakfast was included in the price (£8.50 a night) and there were 8 people to a room.
Everyone was very friendly to us (despite it being a tourist town!) and I would certainly recommend it as a weekend visit to anyone. Probably not for longer than that though as you can pretty much see everything there is in the town itself in a couple of days.
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