Public transit is not very well organized in Lebanon. Collective taxis are the most convenient mode of transport in the city. Between cities, you can take inter-city buses that transit between bus stations.
From the airport to Beirut and its suburbs, your only option is a taxi (which costs close to LBP 40,000 or €20).
Taxis can be identified by their red number plates and the illuminated “TAXI” sign on top of the car.
In cities, you can take a collective taxi (that stops frequently to pick up passengers). In this case, be sure to tell the driver where you are going before getting in.