I would recommend the MRT (train system) which will get you into town in less than 30 mins. It runs regularly and up until late evening.

A map can be found here.

You can buy single use tickets at the station to your chosen destination but you will need Singapore Dollars.

Take the train from the airport (change at Tanah Merah Station) then all the way into Bugis, which is a pretty thriving shopping area with lots to do. It will take ~30 mins by train. You can walk into Little India from there.

Alternatively, go to Outram Park and change to the purple line and go to Little India MRT station.

Taxis are good here but be wary, during peak periods (5-10pm), it is almost impossible to hail one from the street as the demand goes up, and the drivers prefer to wait for dispatched (booked) jobs, where they make extra $$.