The US recently lifted its ban on travel to Lebanon, so direct flights from the US should begin in the near future. Syria is the only country which currently has an open land border with Lebanon - the border with Israel is likely to stay closed for some time. Neither Syria nor Lebanon will issue visas for the other, so make sure you get your visas before you leave home. Buses run between Beirut and Damascus several times a day, and there are also buses and service taxis from Beirut to Aleppo and Homs, or from Tripoli to Lattakia and Homs. 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.