If you're staying in beirut and it's too much hassle for you to take a taxi or a service taxi to the city then book yourself a tour with either wild discovery or nakhal, it'll only cost you around $50 and you get taken to all the major sites plus entry to the castle is included as well as lunch on the highway to tripoli in a lovely seafront restaurant.
Both times I went to Trablous I went from Charles Helou Station in Beirut. There are both smaller buses and bigger buses with air-condition. The bus ride takes about 1,5 hours and the price , when I went, was
1000 LL (in June 2002) and 1500 LL (in August 2002).
To go to Bcharré, minibusses are leaving Trablous from Abdel Hamid Karani Square (near the tourist office). The price is 2500 LL (June 2002).
From Trablous to Aleppo (in Syria) it took about 5 hours.
The most expensive way to get from Tripoli to Beirut, or viceversa, is by private taxi: it costs about 30 dollars. The least expensive is by bus: less than 1 dollar (to be precise 1000 Lebanese pounds). Buses leave regularily between the two cities: in Beirut from Charles Helou station, in Tripoli from the CLock Tower square. I still have the ticket's seller's voice in my ear: Ale Beirut! Aleee Beirut! Aleeeee Beirut! He shouted for 20minutes, until the bus filled up and left.
Buses, taxis and minibuses serve Tarablus from Beirut, although it is hard to find the right station in Beirut. They all take the coastal highway through Jounieh and Byblos (difficult to know where Beirut ends and Jounieh begins, as the coast is very built-up). From Tarablus, service taxis go to Damascus, Homs and Tartus in Syria. The road to Tartus passes a poverty-stricken Palestinian camp on the beach just before the border. This way to SYRIA!! (BTW, the picture shows the cliffs in Beirut)