First of all, you need to forget about the idea that you will have to learn Arabic script writing in order to start Learning Lebanese.
Lebanese people have developed a way of transliterating Lebanese pronuciation into Latin alphabet. Many have invented specific Latin characters. The following web page sums up the transliteration that is mostly used by Lebanese to write over the Internet, and that only uses English letters with no accents.
Use the following site www.abjadiye.com which is the only comprehensive site that uses transliteration with only English accents (no specific keyboard is required).
With Abjadiye.com, you can learn Lebanese in a very user-friendly manner. All words and conversations are illustrated, and grammar rules are provided without too much grammatical jargon.
When using Abjadiye.com you can listen to conversations, and repeat any sentence you wish. You should repeat the conversations as often as you can in order to feel comfortable with the way it is said.
Do the short quizzes Abjadiye.com provides. It will ask you to click on the right word, and to write a missing word in a conversation. It is important you write the words to remember them.
Finally, complete the 99 lessons of Abjadiye.com. This will allow you to start practicing with Lebanese friends so naturally. Do not forget to visit regularly the thousands of nouns, verbs and adjectives provided in the lexicon.
The whole thing will take you around 3 months at the rate of 15 minutes a day.