I would recommend you to walk around downtown, around St. Stephans Cathedral, der Ring, the Rathaus (city hall), the Hofburg.
You will get a great impression of the old Vienna.
If you're a tough customer, you can visit the Narrenturm (Federal Pathologic-anatomical Museum) http://www.narrenturm.at/
We have 2 main places for shopping, the Mariahilferstrasse (located at the Tube Stations U6 "Westbahnhof" and you can walk all the way down to the Tube Station "Volkstheater") and the famous Kärtnerstrasse (huge street from the St. Stephans Cathedral to the Vienna State Opera - Karlsplatz, mostly expensive stuff like designer shops, jewelry and so on).
There is a huge shopping mall, but it's little bit out of town. There is a public transport to the shopping mall, but it takes quite a while to get there. With the car it's not that bad. http://www.scs.at/
Bars/places to hang out:
There are 2 main places:
- Das Bermudadreieck (Bermuda Triangle), located downtown (around the tube station Schwedenplatz), a lot of bars, mostly a young crowd and a lot of drunk people. It's not my cup of coffee but if you want to go crazy, have fun and don't mind a lot of drunken kids, it's the place to go).