There are quite many things to do in Vienna. Obviously you will start with some of the typical attractions of a European city with such great historical heritage:
Visiting / Sight-seeing the historical buildings in the very Inner City may be your starting point. That is easily done by just walking around and you will come by a lot of historical places, notably the St. Stephen's Cathedral, the Hofburg and many historical palais of the former noble Aristocracy of the Austro-Hungarian empire. There are also many art collection in the inner city, one of the most prominent is in the Albertina. If you are into horses visit the famous "Spanische Hofreitschule" (Hofburg). And of course Kaerntnerstrasse and Graben invite you to just walk around or drink coffee in the many cafeterias there.
You should not miss to make a walk around the Ringstrasse (which borders the Inner City). The representative buildings there (Burg Theatre, Opera, Rathaus, Museums, University etc.) were all built during Vienna's so called Gruenderzeit (ca. 1870-1914) where a dynamic urban development process made Vienna to one of the largest cities of this time.
A modern and vibrant art scene is around the Museumsquartier (near the mueseums at the Ringstrasse)
Of course, don't miss Schloss Schönbrunn (the castle where the emperor used to live) and Schloss Belvedere.
If it is a hot summer day you might enjoy an excursion to Vienna's Donauinsel. Inititally a flood protection system by the Danube river it is not a huge recreation area where you can go swimming, sun-bathing, cycling and even water-skiing or wakeboarding.
You also are invited to make excursions to the many beautiful sites around Vienna. If you like mountains visit Reichenau/Rax or Puchberg/Schneeberg, both old towns at the foot of mighty alpine massifs and easily reached by train from Vienna (ca. 1 h train ride). If it is winter, you might even enjoy going skiing (there are many ski resorts around Vienna).
Increasingly attractive among the visitors of Vienna is a boat ride with a high-speed-boat along the Danube to Bratislava, the capital of Slovakia and in itself an interesting city to visit. The ride takes about 1 hour.
Of course, this list could go on almost forever. It is just a matter of taste what kind of activity you prefer. Vienna waits for you, as Billy Joel used to sing.