As noted above, "must do's" are so subjective.
If I had two full days in Vienna, here's what I would do:
Friday - Breakfast at Naschmarkt (Neni's) and then a stroll through the flea market. Stop and look at the Secession and Majolikahaus. Take the U4 to Schonbrunn and spend the afternoon there. Take the U4 back to Karlsplatz and get on the D Tram to Nussdorf and have dinner at Schuebel-Auer heuriger.
Saturday - Follow a self-guided tour through the first district such as the one on Frommer's. This will allow you to see St. Stephan's Cathedral, walk down the Graben, see the Hofburg (Treasury) and wander some side streets. Have ice cream at Zanoni and Zanoni; have coffee and cake at Demel's (upstairs). Eat dinner at Figlmuller's.
For your arrival day, I'd suggest walking around the ring clockwise from Stadtpark to the Rathaus. It is not a long distance and you will see many notable buildings. Have dinner at Palmenhaus.
I have gone to the Prater and rode the Ferris Wheel about a million times (when visitors come) and I fail to see the appeal. That said, Vienna's version of Oktoberfest is now underway at the Prater just in front of the Ferris Wheel. This can create a fun or not fun atmosphere. Fun if you like drinking beer; not fun if you don't like obnoxious drunk people.
I suggest the following English language website for ideas of what to do next weekend:
http://www.viennawurstelstand.com/