In the past, I have done a rail pass which included some days with a rental car. On one of those car days, I picked up a car in Salzburg and went to a castle south of there (I think) which had a great bird show with owls, hawks, falcons, and other raptors. I think it is near the salt mines, which I did not do, but I think you could do both on the same day.
Some of the good stuff in Salzburg is away from old town, requiring a bus ride. Check out Schloss Hellbrunn. Tourist Card was useful. Lots of English-speaking tour opportunities which are easy to find, so don't spend time ahead trying to track those down.