Whoohoooo
I was planning a ride on saturday afternoon, but if you really want I could show you around a bit.....
Otherwise, get a cab from the airport (cheap, easy, metered, and quick) and get it to drop you at the Raffles hotel. Have a quick look round, beautiful place.
Then start walking south down North Bridge Road. (maybe pop into the swisshotel, the massive tower, go up to the 71st floor, amazing views. don't go into the new asia bar, but tell them that's where you're headed. when you come out of the lift just check the view from the window for free, the bar's not so cheap) Have a look round the river area, Clarke Quay, Boat Quay. Lovely views of the skyscrapers, and some beautiful old colonial buildings.
Cross to the south of the river and carry on toward Chinatown, either on eu tong sen street or on south bridge road. Have a good wander around, especially mosque street. maybe grab a bit to eat in food street, loads of choice. Keep going south until you get to Duxton road/Duxton hill. amazing temple at the junction of south bridge road and tanjong pagar road. beautiful old shophouses wonderfully restored in the duxton streets (red light district at night but perfectly family safe in the daytime).
if you still have a good bit of time, maybe get a cab to sentosa island, nice beaches, some cool restaurants and bars, an aquarium for the kids. If you have less time, get a cab to east coast park, it's on the way to the airport, and has a nice beach with some cool breezes.
There are some nice places like the bird park and the botanical gardens, but they are a bit far out and the roads will be busy. You can easily spend all your time around the river and in chinatown. you're never going to see everything, but i reckon those are some of the best bits to tick off the list.
but cabs are cheap and plentiful, so don't worry about getting them, it's not like london in that respect!
i'll write again if i think of anything else