It is important to note the timings of each transport system.
If you don't mind paying a bit more to go around faster and easier, you can take taxi. In Singapore, you cannot hail taxi just anywhere on the streets. You can either wait in bus stops, or in the taxi stands provided almost everywhere (you can easily find them near hotels, malls, and MRT stations).
Be aware, however, that some of the taxi drivers do not really speak English. So it is best if you write down your destination and show them.
The maximum number of people allowed to ride in a taxi in Singapore is four, one at the front seat, and three at the back.
Taxis are available 24 hours, but from 12-6 am, the fees are 50% more expensive.
For more detailed information about taxis in Singapore, including the fares, go here.
Buses in Singapore are both single and double-decker. More and more buses are wheelchair-friendly, so it's not really a problem when you bring a few suitcases, or even a baby stroller (but Singapore is also a highly self-concious society, they are careful in doing things so that they won't bother other people in public, so be considerate).
To pay the bus fees, you can use EZ-link, a card in which you store a sum of money and you just tap when you get into and out of the bus. An amount will be deducted from the card, as the fee.
You can also simply pay by cash, but you'd need to prepare many coins.
The bus fares are approximately between $0.69 to $1.70.
It might get complicated, however, even if buses routes are provided as you might not familiar with the names of the streets in Singapore.
Buses are roughly available from 5.30 am to 12 am. During the weekends (Friday to Sunday midnights), however, Night Riders are provided. They operate roughly from 12-4 am.
MRT (Mass Rapid Transit)
You can either use EZ-link card (there is a kind especially for tourist) or a single-trip card to pay for the MRT fees. You can buy both card from machines provided in the MRT stations.
For the single-trip card, there would be $1 deposit and the trip fee is slightly expensive than EZ-link, so it is recommended that you use the EZ-link if you're staying more than two days and would travel by MRT (and bus) a lot.
The approximate MRT fares you have to pay are between $0.69 to $1.70 (plus $1 as deposit when you use the single-trip card).
MRTs start they operation around 6 am to 11.30 pm.
If you have more time and energy, and prefer more adventure, walking around Singapore can be an exciting option. Although it is not in the top ten list of the safest country in the world, Singapore law is very strict, so you can have no worries of walking almost everywhere, whenever, even during midnight, in Singapore. It is also a very clean country, where you can hardly find trash, but easily find garbage bins.
You can easily get the map of Singapore once you landed in Changi Airport, reach Harborfront, or when you shop around Orchard Rd. Just look out for Singapore Visitors Centers around.
You can also access the map online, here.
Click here to see the future MRT lines.
Click here for online routes generator with Singapore public transports.