Are you driving, Courtney, where are you flying out of, and does the 10 days include the days of arrival and departure? We spent 6 full days based in Vienna on our honeymoon many years ago, after dropping off the car. If I remember correctly, 3 days more or less within the city and catching some of it's main attractions ( I am a big The Third Man fan), a day in the woods and one of the wine villages, a day each in Krems and Melk by train.
I've never been to Budapest. On the way to Salzburg, Steyr is a pretty, working town, and Hallstatt is fabulous in my opinion. You can easily spend 4 days in Salzburg and it's environs, including the Salzkammergut region to the east and maybe a day around Konigssee across the border.