Once you leave Beirut, visit any of the other cities in Lebanon. The country is relatively small, so no single journey to any other city should take more than two hours. Depending on the season visiting, you can go skiing in one the six mountain resorts, or go to the beaches in Tyr or one of the beach resorts in Jounieh. Visit the national heritage sites around Lebanon; every city in the country has at least some form of ancient or historic site.
Buses that head to the North city of Tripoli or to Syria and Jordan operate from the Charles Helou bus station, which is also the station for long-distance taxis that link Beirut to neighboring cities as well as Syria and Jordan. Charles Helou is located next to Port of Beirut. It can be difficult to understand the main bus routes at first but you can always ask, bus drivers are like everybody else in Lebanon, friendly and may be able to communicate in English or French.
Buses that head South -- to Sidon or Tyre -- leave from the Cola intersection.
