How much it should cost and what it actually costs are two vastly different amounts. Anyway, a taxi from the airport costs BZ$50 (also payable in USD, $25) to go to anywhere in Belize City (including the bus terminal). The trip takes about 15 to 25 minutes depending on traffic. Before exiting the airport, try to find another person to share a taxi and split the cost (the taxi drivers won't like it).
There are no public buses from Philip S. W. Goldson International Airport (PGIA).
There was a shuttle service that advertised pickups from the airport, but they were only a few dollars cheaper than a taxi.
The cheap option is to walk or taxi from the airport terminal to the Northern highway (about 2.5 km) and wait for a bus. As you approach the intersection with the highway, Belize City will be to your right so just wait on the side of the highway. When you see a bus coming, wave your arm up and down to get it to stop; Express buses will not stop. If going to Belize City, the fare is BZ$1 or $2, and the bus will terminate at the bus terminal where you can change to a bus to another destination.
There is only one bus terminal in Belize City -- it's still known as the Novelo's terminal (named after a former bus company). Buses to San Ignacio depart from that terminal every 30 minutes.