Vienna, of course - an absolute must! And - I am serious - go to the opera, not the one that looks lige a stack of seashells

DON'T miss several of the most beautiful coffeehouses ("Kaffeehaeuser"), like the "Cafe Central" or, very original, the "Cafe Havelka". Ask for them in Vienna, they are known.
Then, try Salzburg with it's castle and a wonderful old town. And if you can make it to the "Hohen Tauern" a mountain range with peaks close to 12000feet, try to go hiking there. The wine from the area around the Danube river ("Wachau" area) is very good, too.
Have fun, it is a gorgeous country!