All you need to do is to head to tourist customer service counter at any CentralWorld Mall and sign up for a tourist pack. The tourist pack includes various coupons and more importantly -
a voucher for a free traveller sim card.
This AIS SIM card offers 10MB internet and 50 minutes local calls.
AIS has another really cheap plan for traveller - a SIM costs 199 bahts and comes with 7 days unlimited data, free calls to other traveller SIMs and 15 bahts call credit.
If you are travelling with your friends/ family, each person can get one tourist pack and it can make your trip much easier. As I did not know about it, I bought my SIM card before going to CentralWorld.