The first thing: do you need to go into Belize City for any reason? If not, then you can avoid it.
As you drive out of the airport to the Northern Highway, turning RIGHT onto the Northern Highway will take you to Belize City.
If you don't need to go to Belize City, then turn LEFT onto the Highway and go for about 5 miles. You'll reach a T-intersection, and there will be a police checkpoint at the intersection. Turn left onto the Boom Road and go for about 12 miles (passing the prison) until you come to a roundabout/traffic circle. Turn right onto the Western Highway and go until you reach the turn off to Belmopan. There are signs at all those intersections.
That part of the trip should take about 1 hour and all the roads are paved. You can travel at the maximum speed limit (50 mph) for most of the trip.
The road you turn onto from the Western Highway is the Hummingbird Highway. Go for about 1.5 miles and you'll reach a roundabout. Go straight for Dangriga or turn left if you want to see our capital. There are signs at the roundabout.
The Hummingbird Highway is winding in parts and very scenic. Maybe 1.5 to 2 hours to reach Dangriga.
If it's raining, the Western Highway gets VERY slippery.
Check it out on Google Earth.