There are no public buses to/from Philip S. W. Goldson International Airport (PGIA). Taxi fare to Belize City is BZ$50.00 (or US$25) per taxi (not per passenger), and takes 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 cheap option is to walk or taxi from the airport terminal to the Philip Goldson Highway (about 2.5 km) and wait for a bus. Upon reaching the T-intersection with the highway, Belize City will be to the right, so just wait anywhere on the side of the highway. Orange Walk, Corozal and Mexico to the left – if going there, cross to the other side of the highway and wait. When a bus approaches, wave an arm up and down to get it to stop. Express buses will not stop. The fare to Belize City is BZ$1 or $2, and the bus terminates at the bus terminal where you can change to a bus to another destination.
For a cheap way to get back to the airport, get a bus to the airport turnoff in Ladyville, and either walk the 2.5 km to the terminal, or look around for a taxi. Bus fare from Belize City is BZ$1 or $2 to the turn-off, and the taxi probably won’t be more than BZ$15.
Google "BelizeBus" for a website all about transportation in Belize.