Singapore's public transport is cheap, extremely efficient and makes getting around the island a breeze. Singapore is undoubtedly the easiest city in South-East Asia to get around.
We don't use taxis in Singapore very much, since MRT, LRT (Light Rail Transit) and busses are cheap, frequent and they can get you almost anywhere.
If you are going to stay in Singapore for a longer period of time or if you frequently travel there, get yourself an EZLink card (a contact-less credit-card sized plastic ticket with stored value inside) and use it for riding all the trains and buses. Fares are cheaper when using EZlink card, but you have to pay a SGD 5 non-refundable card fee when buying your card.
Getting from the airport
From Changi airport, the cheapest and very convenient way to the city centre is by MRT. You need to transit at least once (at Tanah Merah MRT station), thus MRT is good for backpackers, but if you have heavy suitcases, travelling by MRT is only OK if they are on the wheels. Else, take a taxi. It shouldn't cost more than SGD 20 to the city centre.
Buses
Bus fares in Singapore are pretty cheap. Bus drivers do not return change, so prepare the exact amount (or use EZLink card). Bus numbers and routes are clearly displayed at most bus stops. If you need assistance with where to get off, ask the driver. Most of them are friendly and willing to help. You can plan your bus route online at Singapore's bus company website.
Mass Rapid Transit (MRT)
MRT is Singaporean futuristic metro system. Singapore's North-East Line (NEL) was the first fully computer-driven heavy rail system in the world. In other words, NEL's trains don't have drivers. Fares are very reasonbable. You can get practically anywhere in Singapore for less than 2 Euros. Tickets are obtainable from machines at the stations. Do not throw away your single MRT ticket, as you get back a SGD 1 deposit when returning it to the machine. Trains generally run from 5.11 am till 12.26 am. Each MRT station has a 'Control Station'. Officers working there would be most happy to assist you if you have any problems with your ticket, or give you directions.
Taxis
Taxis in Singapore are cheap and there is a lot of them. Major companies are Comfort and CityCab. You can get a cab at taxi stations near bus interchanges, MRT stations and shopping centres or you can just stretch out your hand to flag one down virtually anywhere else. Don't do it at Orchard Road though or at any other major roads. It is illegal to stop there.