What I can recommend you to do if you have only one full day in Vienna is the following:
8.30h Stand up, get yourself a nice sandwich from one of the many bakeries.
9.00h Take a tram, metro, bus to the city center. Start with taking a tram or a guided tour along the rings: Schottenring, Burgring and Opernring. You will see many of the nice buildings Vienna has, like the Votivkirche, the main Vienna University building, the Rathaus, the Burgtheater, the Parlement, Museumsquartier with the Naturhistorisches and Kunsthistorisches museum, Hofburg, and the Opern building.
10.00h Get out at the Opern building and take a walk along the shopping street Kärtnerstrasse to the Stephansdom.
10.30h After the Stephansdom you can go for a coffee at one of the many coffee houses Vienna still has, for example Hawelka.
11.30h Go take a walk through the inner city. Make sure you take a look at Albertina, Hofburg and the Spanische Hofreitschule. Parks are also recommendable, for example the Volkspark.
13.00h Time for lunch. Many restaurants have special lunch offers with buffets for reasonable prices.
14.30h Time to leave the city-center behind and travel to Schönbrunn, Prater or Belvedere. All three worth it to visit. If you only have one day I would choose Schönbrunn, because it is really beautiful with the surrounding gardens and zoo.
17.30h Travel back to the Mariahilferstrasse, the big shopping street. Find “Il Mare” and have a nice pizza for dinner.
19.00h Make sure you are in line for a ticket for an opera performance. A must when in Vienna.