The Salzach river is a big part of the city and can be quite pretty. HOWEVER, the part down near the Old Town and Getreidegasse is very loud and crowded and there are busy roads hugging the river on each side.
Keep walking north along the river, towards the train tracks. Then you'll get to the pretty, peaceful part of the river. Around the Mullnersteg pedestrian bridge (you can see this on Google maps).
Was near the head of the Salzach on the alte Gerlosstraße/Krimmler Landesstraße (B165) this summer. Pretty peaceful there. We stayed in a larger modern hotel once in central Salzburg and walked over the Müllnersteg and along the Salzach as our route into the the old town.
My first visit to Salzburg, I limited my visit to the Old Town area and the area directly across the river. On my second trip, I discovered the area mentioned by phred. It really is a nice area for walking. Of course, my suggestion is to keep walking north, go under the train tracks, ascend the stairs to the raised road that crosses the river and walk through a bit more gritty part of the city to reach Salzburg's library in a not-so-gritty area. The library has cool architecture and a nice reading room/cafe at the top with nice views of Salzburg, including the fortress in the distance.
Phred is spot on. Bonus to that area, you're also very near the lovely Closter Mulln Augustiner beer garden. It's the area I like to stay when I'm in Salzburg.
