There are two cell phone providers here in Belize, the one (Digicell) uses conventional GSM phones with separate SIM cards. These are readily available from BTL (the service provider) at any of their offices. There is one in San Pedro at the north end of town. Some other shops sell them and it may well be possible to buy one at the airport, but you'll need to ask once there.
There is also another provider, SMART, which uses phones that are hard wired for the number and don't use a SIM card (can't remember what this type of service is called). There's no point in buying such a phone specially, but they can be rented on the island for a small amount from Quicksilver. I'll post contact info if you want - don't know it offhand.
SMART provides generally better coverage than Digicell, is a bit cheaper per minute, and after the first minute charges by the second (Digicell charges by the minute). But they both have their place and I carry both all the time.
An alternative if you're bringing a laptop is to use Skype. Until very recently that needed a VPN to work, but apparently the block placed by BTL on it (all telecommunications pass through BTL networks, including all internet except satellite and SMART phones) has been lifted, and it now works without a VPN. Damn good idea to have a VPN anyway mind, if you value your privacy. Best to use a compact headset along with your own laptop. Free wireless internet is widely available all over the island. Basic Skype allows you to make calls but not to receive them (from a normal phone). However, for a bit extra you can have a phone number which can be called. The way I use Skype is in conjunction with a cellphone - if it's going to be a long call the caller sets it up by calling me on my cellphone first, then I call them back on Skype. Or at most of the internet cafes you can make VoIP calls, though not desperately cheaply and of course without much privacy.
You may prefer to use your home SIM card so people can call you, then tell them you'll call them back and use Skype for that. Could save you a lot of money. I can't believe they'll want to make an international call to you, and this way they won't have to. But don't make calls from Belize with your home SIM card unless you have a very large bank balance you want to give away.