One of the first things that you should get is the EZ-Link card, which allows to use both buses and the MRT trains without having to buy tickets all the time. You can buy it in any MRT-station, and you'll keep the card for your entire stay (it is rechargeable). The best thing about the card is that you can keep it in your wallet, even when you hold it against the card reader!
Singapore has a very clean and modern subway system, which is called MRT. Travel on the trains is relatively fast and inexpensive (with the most expensive fare at S$2.60 or so, but you'll never really pay more than S$1.80). Always check when the last train leaves: On weekdays this can easily be shortly before midnight.
As the MRT does not cover every corner of the island yet, you might have to resort to buses every once in a while. There are currently two bus companies that take you everywhere, but really the bus system is rather confusing. If you use the EZ-Link card, you have to use it when you get on the bus, and if you want to save some money, you use it again when you exit.
Your third option to get around is the taxi. Taxis are obviously the most expensive means of transport - if you travel alone. If you are in a group of 2 to 4 people, a taxi might in fact be cheaper than the MRT! Watch out for some surcharges that you might not expect: from midnight to 6am you pay 50% more than what the meter shows (minimum S$3.20), from the airport there is a surcharge between S$3 and S$5 depending on when you go, entering the CBD costs S$2 at the busiest times, and there are usually surcharges for phone bookings, unless you're lucky to find a taxi company that offers you a temporary deal for free bookings. Taxi rides do not normally cost very much - from the Beach Road area to the airport or to Yishun MRT you pay between S$14 and S$15 - check the map and you'll know that your regular trips will be much cheaper. Expect to pay between S$3 and S$7 for a 5-10 minute ride.