One day in Vienna is enough to see its imperial core if you move efficiently and accept that the outer palaces, smaller museums, and lesser churches will have to wait for a return trip. This route sticks to the historic center and one essential palace, skipping everything else on purpose.
Start early at Schönbrunn Palace before the tour buses arrive, walking the gardens if time allows but keeping the interior visit brisk. Head back into the center and spend the late morning at the Hofburg Imperial Palace, the seat of the Habsburgs, then step outside to admire the grand Ringstrasse buildings nearby, including the Rathaus and the Parliament Building. For lunch, cut through to Graben and Kärntner Strasse, Vienna's two most elegant pedestrian shopping streets, and pop into Peterskirche along the way for a quick look at its Baroque interior. In the early afternoon, cross to St. Stephen's Cathedral, the city's Gothic centerpiece, and climb or simply circle it before wandering the medieval square of Am Hof. Spend the mid-afternoon at the Kunsthistorisches Museum for a condensed look at its old master paintings, or swap it for the Albertina if you prefer graphic art and a shorter visit. As evening approaches, make your way to Karlskirche to see its dome from the reflecting pool outside, then finish the day near the Vienna State Opera, catching a standing-room performance if tickets are available or simply enjoying the illuminated facade.