Must do/see suggestions for Vienna?

Depends what you like--if it's art, then Vienna is full of it, particularly the Secession. Beautiful churches, dancing horses, opera, choir, palaces...My must-dos are most probably not yours, my trip was all about sites pertaining to The Third Man--Harry Lime door, the wonderful museum, walking tour, ride on the Riesenrad, various other movie locales, with a lot of Secession thrown in for a perfectly wonderful trip.

I agree with Christa - it depends on what you like. As an art and opera buff, Vienna was pure heaven. We spent half a day in the Kunsthistorische Museum, but also loved the Belvedere and the Albertina. Ran out of time and didn't make it to the Secession. I much preferred the Hofburg Treasury to the Sissi Museum. Watching the Spanish Riding School morning exercises for a few minutes was enough for us.

If I had to do one thing over regarding Vienna, I would have paid for more expensive seats at the opera.

Definitely the Kunsthistoriches. A stop at Julius Meinl for coffee and packaged food goodies to take home. The Leopold if you like modern art. A very controversial Schiele is there, but it's not the content that makes it so scandalous: The Portrait of Wally Neuzil was a looted painting that the government of Austria wouldn't reimburse the estate of the owner for until 2010, when they were found to be complicit in its theft.

I thought everybody had to go to Schoenbrunn Palace. Spectacular. And at least see the outside of the Opera House since it pops up in movies all the time.

I happened to be staying on Operngasse, right across from the Opera, in August of 2014 when they were filming one of the Mission: Impossible movies and it caused plenty of disruptions, especially the night Tom Cruise was dangling off the roof of the Opera.