When you land - go straight to the tourist information desk. Like Gemuser said they have loads of free options.
You can get a free pass for 10 rides on the mass rail transit (subway) system, or you can get a free return coach ride into Singapore, you can have a free tour of Singapore as well.
With that you get a pass you can use for either a free pizza and drink at the airport, a shower or a free meal somewhere else in the airport.
When you get downtown, you have to do the Duck Tour - a ride on an ex-Vietnam amphibious vehicle which takes you around town then into the harbour, very good fun, http://www.ducktours.com.sg
Also for shopping you'll want Orchard Road - lots of duty free shops and you can get a taxi there from anywhere in town, or a ride on the subway.
Hope this helps!