Traveling to Vienna, Austria? In my opinion, the top 2 sites you would have to see are the Schönbrunn Palace and the Hofburg Palace. These two palaces were the imperial residencies of the Habsburg Monarchy until the end of World War I. The Hofburg Palace was Monarchy's primary residence during the winter months and is currently being used as a museum, concert hall, banquets,etc. A must see within the museum is the imperial silver collection. This collection has a variety of the royals diningware, tables, silverware, etc. My personal favorite is the Milan Table Centerpiece, which can extend to up to a length of 30 meters!
After walking through the Museum and taking everything in, I would check what time the next concert would be (typically at night). I must admit that my knowledge of classical music is limited, but Vienna being the home of Mozart, makes the experience very memorable.
The Schönbrunn Palace was the summer residence of the Habsburg Monarchy. It offers awe-inspiring gardens along with an extremely expansive residence. I would recommend going during the summer while the gardens are in full bloom; however, winter does offer some very scenic photography.
On your adventure to Vienna, Austria, I would definitely suggest taking the palace tour, highlighting the study of Emperor Franz Joseph, the family dining room, the Small Gallery and the Great Gallery (ceiling paintings by Gregorio Guglielmi). It sounds like a lot to take in, but it's all completely worth it.
Again, Make sure to purchase your tickets in advance to save time!