As one of the most prosperous and modern countries in the world, finding your way around Singapore is not much of a problem. It is also affordable and therefore there is little need for extravagance in hiring your own car, unless you want to of course.
The most common way people move around the city is by rail, called the Mass Rapid Transit (MRT) and the Light Transit Rail (LRT). These days, tickets cost no more than $2 for a single ride and become incrementally cheaper the closer you plan on traveling from the departure point. If you are planning on staying in Singapore for some amount of time, perhaps the best thing to do is to purchase an EZ-Link or Nets FlashPay card, that way you do not have to limit yourself to seeing just a few places.
Another mode of transport that is popular amongst the public, albeit less popular than the rail, includes the bus. Although the bus is a bit more complicated than the rail, it is a great option if you would like to get a better feel for the city, since it is not underground. As with the rails, the fare card options above are convenient in that they also work here.
If you have a little more to spend, the taxis in Singapore are pretty trustworthy and are priced very reasonably. Since they work on a meter, you can at least see what you’re paying and need not about negotiating a price. As a safe-marker, inner trips should not cost you any more than $10.
Finally, if the normal means of transport are not as exotic as you had hoped, you can always hop on the trishaw, located around the Singapore River and Singapore’s Chinatown. Mind you however that these are really just for short trips and sightseeing, so avoid any real treks across town in one.