Understand the differences and similarities between Lebanese Arabic and standard Arabic. Many words are identical, but a great many have changed or are pronounced differently. They can be mutually understood, yet remain unique languages. Lebanese Arabic is generally a simpler language.
Learn the standard Arabic alphabet and writing system. This is an optional step but an important one for people wishing to become fluent in Lebanese Arabic and be able to communicate with it.
Study Lebanese online. Start with free websites like Wikitravel Phrasebook or look for podcasts or blogs that teach Lebanese Arabic. There are a handful of resources online.
Learn Lebanese with a tutor or professor. If you live in a large city, look for someone through Craigslist or by posting a notice around a large university. If you can't find a Lebanese tutor in your area, consider getting an online tutor whom you can talk to with Skype or other audio chat services.
Study Lebanese in Lebanon. The best way to learn a language is to be immersed in it. Study at the American University of Beirut or Beirut Arab University. Or just go there and learn by living.