I spent a week in Vienna in late August 2014 and with a couple of day trips it was the perfect amount of time to get comfortable and explore the city at a pleasant pace. I love opera but it wasn't in season--though when I was there they were filming the exterior scenes for Mission: Impossible-Rogue Nation and I did see Tom Cruise dangling from the opera house.
My must-dos were different and I spent a lot of time on things to do with The Third Man--finding filming locations, walking tour, riding the Riesenrad, visiting the wonderful museum devoted to it (and post-war Vienna, very interesting)and even seeing it at the theatre near my hotel. I love the art of the Secession era so tailored my museum agenda to that, also enjoyed the area near the Hundertwasser haus. I planned to be there on a Saturday so I could visit the flohmarkt at the end of the Naschmarkt, very crowded but I found some great old things so very much worth it. I spent a morning at Schonbrunn and also toured parts of the Hofburg. It's a great city for walking, the Ringstrasse is very compact but I also made good use of my pass and rode the U-bahn and trams.
I did a day trip to Melk--train there, toured the Abbey and had lunch in the charming little town, then boat along the Danube to Krems where I caught the train back to Vienna. My other day trip was a very long one to Salzburg--3 hours each way, left at 6:30 am and back in Vienna by 8:30 pm. It's do-able if you plan what you want to see--I used a HoHo bus to get around to the outlying sites, toured Hellbrunn palace and gardens, ended in the old town where I toured Mozart's gebursthaus and spent the rest of the day exploring, having lunch and finding a craft fair along the river.
I loved Vienna quite a lot and will be returning for 3 days next April--missed the Kunsthistoriche and will re-visit the Belvedere and Leopold, and just spend time exploring.