Salzburg vs. Innsbruck or anywhere else?

Vienna is another great European city, and one we love--The Music City of Europe. It is a minimum 4 day place, preferably more.
Bratislava is only 39 miles and a one hour bus ride from Vienna. It is a very modern, high tech city and can be a day trip. We spent a weekend there recovering from a week of hard touring.
Salzburg is another very popular Austrian city easily reached by train from Vienna and Munich.
I spent a summer in Innsbruck in college, so I am naturally drawn back to the incredible beauty of The Alps 2 hours south of Salzburg. We find Tirol to be simpler and much cheaper to travel through than Switzerland. It is also the gateway into Northern Italy.

I would say at least three nights in Vienna. Salzburg is lovely and I would say at least two night. I haven't been to Innsbruck, mostly because people have told me that other cities are so much more worthwhile. Munich is a bit over an hour from Salzburg if that might be of interest.

There's history, and then there's ... history. I was struck, during a cemetery walk, to find that the American Military Governor had come to love Salzburg so much that he and his wife are buried there! Both Salzburg and Innsbruck are lovely places, although Salzburg is a bit dominated by Classical Music history. Vienna is a vast city, and perhaps expensive for a student budget. You may want to plan in advance for tickets that are important to you, like the Spanish Riding School or something like that.

Presumably you already know that August is the Euro vacation month, and along with foreign visitors, things can be very crowded and also hot.