Lots to see and do in Vienna, one of our favorite cities! Where are you staying? If you are in the downtown area I recommend taking the S-Bahn from the airport, half the price of the CAT train and just as easy.
http://www.viennaairport.com/en/passengers/arrival__parking/s-bahn__suburban_railway
The inner city (inside the Ring and even just beyond to the museums) is very walkable. But you may want to consider a transit pass to use the trams.
http://www.viennaairport.com/en/passengers/arrival__parking/s-bahn__suburban_railway
Here is a great website for all kinds of Vienna info:
http://www.wien.info/en/sightseeing
Our favorite sites inside or walkable near the Ring include:
St. Stephens´s Cathedral (Stephansdom)
Imperial Palace - Hofburg
Belvedere (Klimt museum)
Kunsthistorisches Museum (Art Museum)
Imperial Crypt (beneath the Capuchin Church)
Try to see a concert at the Grand Hall of the Musikverein (where they hold the famous New Year’s Concert). We did the really touristy Mozart Concert there (which is looked down upon by many classical music aficionados) but we loved it. Just sitting in box seats in that beautiful hall was worth every penny.
We also liked the touristy Waltz King, Johann Strauss concert (held in different locations around town).
If you like Opera you’ll probably want to check out the schedule at the Opera House (or just take the guided tour of the hall, it’s really beautiful).
Other music related sites are the Mozart House museum, the Beethoven House museum and the House of Music Museum is excellent.
And many smaller churches in the center, several have free concerts (organ or choir), check the tourist brochures when you get there for current schedules.
We never had the chance to hear the Boys Choir (or see the Lippizaner Horses) but hope to do that someday.
Outside the city center the must see is the Schönbrunn Palace (easy to get to on public transport) and the Prater where the famous Ferris Wheel is (rent the movie The Third Man before you go!).
Stop in at the many coffee houses, all are different, very historic and beautiful. Two of the restaurants we liked on our last trip (both inside the Ring) were Fratelli (Italian) and the 1516 Brew Pub.