All of Belize's towns, including the parts of Belize City that most visitors frequent, are small enough to cover on foot, although for safety reasons you should take taxis for some trips within Belize City. Taxis are plentiful in all mainland towns and are also an option for getting to places out of town, although asking taxis to venture beyond their normal work area can be fairly expensive.
Bicycling is an enjoyable way of getting around local areas and bikes can be rented at around BZ$20 per day in many tourist haunts (and are free for guests at some accommodations). On the cayes, of course, you get around by boat if you're going anywhere offshore.
If you're spending some time at the beach and you can't fathom being dependent on your own leg power, you might consider renting a golf cart. It's relatively inexpensive (compared to a car) but it still gets you to the beach and back without causing you to break a sweat. The golf cart is a popular form of transportation in Placencia, San Pedro and – to a lesser degree – Caye Caulker. Both gas-powered and battery-powered golf carts are available: gas goes further and faster, but battery is better for the planet. Expect to pay about BZ$100 to BZ$130 per day for a four-seater.
Hitchhiking is never entirely safe in any country and in Belize, like anywhere, it's imperative that you listen to your instincts and travel smart. Travelers who decide to hitchhike should understand that they are taking a small but potentially serious risk. You're far better off traveling with another person, and never hitchhike at night. Also keep in mind that buses in Belize are cheap and fairly efficient; you might decide that a bus is a safer and more comfortable bet.
Hitchhiking is a fairly common way for Belizeans to get around. In a country where vehicle owners are a minority and public transportation is infrequent to places off the main roads, it's common to see people trying to catch a lift at bus stops or at speed bumps, where traffic slows down. If you too are trying to get some place where there's no bus for the next three hours, it's likely that you'll soon get a ride if you hold out your hand and look friendly. Offering to pay a share of the fuel costs at the end of your ride never goes amiss. But always be aware of the potential risks.