Salt Mines: They can be a bit pricey but the reviews are great. You get to ride down into a salt mine and explore and learn about one of Salzburg's (salz meaning salt) most important resources.
Eagle's Nest: This is one of Hitler's residential escapes. Although he did not go there often because it was high on the mountains and he was scared of heights (this home was given to him) there is an awesome and cheap documentation center about Austrian-German history. Eagle's Nest is actually in Germany but the Austrian German boarder is super super close in Salzburg, in fact you probably won't realize you have crossed the boarder! It has a great documentation center that has English, and a bunker. It is a must if you go near Munich or Salzburg.
Mozart Concert: Salzburg is the city of Mozart. He was born there and Salzburg is super proud of him (even though he hated Salzburg growing up!) Try to go to a Mozart concert, it will astound you. There is a Mozart festival the end of January that is amazing, and tickets are reasonably priced. Be sure to dress up tho and be on time. Austrians do not tolerate being late!
The Fortress: It is an icon of Salzburg and holds a lot of history. You can ride a cable car up for I think 7 euros or take the hiking trail up and explore the wilderness and fortress walls before coming to the fortress.
The Monastery: There is a beautiful yellow monastery on the other side of the Salzach (main river) across from the fortress that gives one one of the best views of Salzburg. It is a great little hike and very interesting. And free!
Mirabell Gardens: The beautiful Salzburg garden with fountains and beautiful flowers! It is wonderful in the crisp winter snow or when it is full of color in the summer. I always loved the old people sitting and chatting on the benches. :) Plus the fountain that Maria sings I have confidence around in the Sound of Music is in Mirabell garden. It has the horse in it.
The Sound of Music: Austria is known for the Sound of Music - but Austrians hate the movie. So do not bring it up with an Austrian and do not go down the streets singing to songs. Not the way to make friends. But you can definitely pay for a tour our hunt most of the sights down by yourself for free. They do appreciate the income the movie brings.
Haus de Nature: An awesome interactive nature museum that is right across the main bridge on your right hand side on the fortress side of the Salzach. I believe it is 7 euros to get in, but it is so fun, even for adults!