I've made the airport <----> Placencia drive a number of times. Three hours sounds about right.
The car rental offices are right across the parking lot from the airport, about a one minute walk. There aren't many roads in Belize, so getting around by car is actually very easy. The roads you'll be on aren't great but they will be paved the entire way. And they're actually in decent shape for the most part except for a couple of quick stretches on the Western Highway which will get a little bumpy but nothing too serious. And you could divert off on to a dirt road (the Coastal Highway) for about 60 miles, but I don't recommend it for first-timers and a four wheel drive vehicle would be a really good idea. All the roads, except for that dirt one, will be passable even after a big rain. If you do get caught in a torrential downpour, find a place to pull off the road and wait it out. You probably won't have to wait for long.
The most scenic part will be the Hummingbird Highway, which begins in Belmopan, the capital "city," and goes all the way to Dangriga on the coast, but you'll turn off on the Southern Highway about 10 miles before you would get to Dangriga. The Western Highway is mostly wet savannah country until you hit the hill with the zoo, then it turns into higher, somewhat dryer scrub that becomes more jungle as you go along. The Hummingbird cuts through the Maya Mountains and is absolutely beautiful. There will be a number of one lane bridges on the Hummingbird so be prepared to wait your turn if there's a line on the other side. The Southern Highway is mostly flat to rolling hills with orange groves and banana farms. And the road after the turnoff to Placencia is US quality except for a really rough patch on a curve just after you reach Riversdale at the top of the peninsula.
Don't expect the rental car to be up to the standards here in the US. You're probably going to get something a bit um, scruffier. As long as the brakes work, you'll be fine, lol.
One thing to be very aware of is how Belize controls car traffic as you pass through a village. You're unlikely to see a cop except at the checkpoint on the way to Belmopan or possibly the one on the Hummingbird Highway. Just as likely they'll be manned by Belizean soldiers. Show them your driver's license, and they'll wave you through. So instead of traffic cops, they use speed bumps. Seriously big speed bumps. As you enter a village, the typical pattern is a set of three warning speed bumps that you can hit at speed with no problem. They're just telling you to slow down. Then the real speed bumps start. They are usually marked with a sign but not always, so keep a sharp lookout. All it takes is once, unless you're a slow learner like me, ahem, in which case it might take twice, for you to be launched into the air by one of these things. There's usually at least two or three of these placed at random in a village. Sometimes, they're marked as a pedestrian crosswalk, which is pretty hilarious as there's no reason why a villager would cross there as opposed to anywhere else. Then you cross the warning speed bumps again and you're on your way. The first ten miles or so of the road into Placencia is wide open, but once you hit the first warning bumps there will a lot of speed bumps the rest of the way down the peninsula.
A couple of highlights: the Belize Zoo is on the Western Highway about halfway between Belize City and Belmopan. Big sign on the right as you climb the first real hill. It's a very cool zoo run by an ex-pat, Sharon Matola. Just past the zoo are a couple of American style restaurants on the left if you want to stop for lunch. Soon after leaving Belmopan on the Hummingbird, you'll see a Mennonite dairy farm on the left. They sell milk and ice cream. Great place to stop for an ice cream cone. Just before you get to MP 30 there's the locally famous "spot" where you can stop, put it in neutral and roll backwards uphill. It's not marked, for obvious reasons, so you would probably need a local to show you but if you see the marker on the left, you could turn around, go back to the top of the hill, then start coming back down and see if you can find it. It's really best, though, if you do this at night so you can see traffic coming from a long ways away. Further down the road, you'll see the signs for The Blue Hole. Nice place for a dip. After you come out of the mountains there's a fruit processing plant on the left where they process all the fruit. They have a small store at the entrance gate. Great place to grab a cold fresh juice.
Be prepared for high gas prices. Typically double what we pay in the US. Gas stations in Ladyville, Belmopan and just before you turn on the Southern Highway.
You have two choices when you leave the airport: turn right, drive the 15 or so miles into the outskirts of Belize City, look for the signs that direct you to the Western Highway to Belmopan and go that way. Or turn left, drive through Ladyville, then turn left onto Burrell Boom Cut and hook up with the Western Highway in Hattieville. I always turn left, but I know where I'm going. Still, I hesitate to send you into Belize City so I recommend that you get a good map or at least directions and turn left when you leave the airport. The turn on Burrell Road isn't terribly well marked but it's the only road of consequence off to the left. Takes about 15 minutes to get there from when you first turn left leaving the airport. If you see signs for the Baboon Sanctuary, you're headed the right way. Other than that, you should have no problems finding your way around.
This probably won't be an issue, but driving at night on the highways is not a pleasant experience. Street lights are pretty much non-existent and so are sidewalks, so the locals like to walk out on the roads. Try to plan your trip so that you're not driving at night as much as possible.
