If you're planning to drive into Lebanon, be prepared to pay a hefty fee at the border - it's refundable when you leave, but it can be very steep. Most people use service taxis to get around, a huge number of which run like buses on set routes. They carry around 5 passengers, each of whom chip in for a fifth of the fare. There are also many 'pirate taxis' cruising for fares. These are more expensive than service taxis, but look exactly the same, so it's best to ask before you get in.
Buses travel between Beirut and other major towns, but service is infrequent and un-timetabled.
There are plans to restore the country's inter-city bus service, but those plans are still on the drawing board. It's far less likely that the country's rail service will ever be restored. Car rentals are fairly expensive in Lebanon and the country is notorious for the bad condition of its roads and the hair-raising style of its drivers. Road rules are effectively non-existent, traffic jams are ubiquitous and there are no speed limits. On the up-side, in theory everybody has agreed to drive on the right, and fuel is cheap and easy to get.