Budget (2015): Expect to pay at least $15-20 per night for a dormitory bed. Some places offer free breakfast, so take advantage of that if you can. If you stay away from nice restaurants, you can easily eat for under $10 per day. Because of the country's size transportation is quite cheap. Minibuses (the common method of intercity travel) are usually no more than $1.50 for a one way trip. Use shared (Service) taxis if you need to move around within a city, as they are cheaper than regular taxis. If you go to Lebanon with a budget mindset, you can manage on $35 per day. However, if you want to go even a small step up in luxury (lots of beer, clubs, private rooms, etc), it can be a very expensive country.
Note: Israeli passports holders are not permitted to enter Lebanon. If you have a second passport (from another country) it should be no problem. Also, if you have an Israeli stamp in your passport, or any other indication that you have traveled to or will travel to Israel, you'll be denied entry. If possible, go there before you enter Israel. If you already have an Israeli stamp, you will need to get a new passport. Although Israel doesn't stamp passports at the airport, they do stamp at land border crossings.
This is the capital city of Lebanon, and boy is it interesting. It feels simultaneously "lived-in" and modern, "old-fashioned" and trendy. There are bullet holes all over the place from the country's civil war, next to sparkling-new Dunkin Donuts' and shopping malls. Beirut is a big party city, so you will find numerous bars crammed into small areas. It is a complex and inviting city, one that has plenty of nooks and crannies to explore.
A UNESCO World Heritage site, Byblos is chock-full of ancient ruins and souvenir shops. It has a nice waterfront, and makes for a relaxing, albeit "touristy" day. Fun fact, it is among the oldest continually inhabited cities on Earth, and the word "Bible" originates from the city's name.
Chances are, you're visiting this city for the Jeita Grotto. For $12, you get to walk through two beautiful caves. The limestone formations are quite pretty.
Also called "Saida," Sidon is Lebaon's third-largest city, located alongside the Mediterranean coastline. It has a well-known ancient sea-castle, which can be visited for under $3. The city itself is nice to walk around, especially alongside the water.
Also called "Sour," Tyre is located in the south of Lebanon, also along the Mediterranean coast. Like most other tourist cities in the countries, it has plenty of ancient ruins for the enterprising history lover.