Train and bus are the most common ways to travel around Belarus. Bus is a little faster and cheaper, but trains are generally more comfortable.
Hitching is practised by young locals quite a bit. But it's never entirely safe in any country in the world, and Lonely Planet doesn't recommend it.
It's perfectly possible to hire a car in Minsk, and, if you can read Cyrillic, it's surprisingly easy to drive in Belarus. In general cars are old and badly maintained. Look them over carefully and check the spare tyre before you drive off. The Brest-Minsk highway (Brestskoye shosse; E30/M1) is an excellent two-laner, but there are frequent tollbooths (they only charge cars with foreign licence plates).
Drivers from the USA or EU can use their own country's driving licence for six months. Cars drive in the right-hand lane, children 12 and under must sit in a back seat, and your blood-alcohol level should be 0%. Fuel is usually not hard to find, but try to keep your tank full, and it would even be wise to keep some spare fuel as well.
You will be instructed by signs to slow down when approaching GAI (road police) stations, and not doing so is a sure-fire way to get a substantial fine. You may see GAI signs in Russian (ГАИ) or in Belarusian (ДАЙ).