I didn't get a chance to go to the State Opera House, but I think that would be the best advice I could give to anyone planning to travel - plan to go! And plan ahead! Otherwise, watching it from outside on the plaza is quite an excellent experience.
Karlskirche is one of the most beautiful churches I've been in. It reminds me of the Sant'Agnese in Agone by the Piazza Navona in Rome. The finishes are just beautiful, and you can head to the top!
This fantastic palace has awesome paintings with two collections: upper and lower. The upper Belvedere has quite the collection of Klimt pieces, and on the way to upper Belvedere is a gorgeous garden.
Between the Rings
Rathaus, Parlament, the museums flanking Maria-Theresien-Platz, and the Opera House all exist in between the inner and outer ring circling the center of the city. This walk takes you through Rathausplatz and Volksgarten, which are both beautiful parks.
Itching to Get Out
Semmering is a great half day trip outside of the city, which gives you a flavor of the Austrian "hills". There is mountain biking in the summer too! But the real beauty is the vast landscape that you get to experience from the top of the lookout tower at Semmering. Quite worth the trip. The Liechtenstein house was also a delight! The apple pie that we had there was on point.
LE SHOP had the Design Guide Vienna, which gives a flavor of each of the districts and where to eat and shop. Highly recommended as a guide, although I'd say that a lot of the districts are pricey, especially the 6th and 7th districts. Still, worth it!
Wiener Schnitzel and Tafelspitz are the two most excellent traditional Austrian dishes, and Plachuttas Gasthaus zur Oper is the place to go!