Check to see if your local community college, university or Arabic cultural center is offering a class in modern standard Arabic. This will give you a basis in the written and spoken language. Once you have the basics you can work on your spoken Libyan.
Ask friends and people at the Arabic cultural center if they know anyone from Libya who might be willing to teach you the specific dialect. If you can find a native speaker, you can try to get your Libyan accent right from the beginning.
Learn the Arabic alphabet. Like English, Arabic has a symbol for each sound. Arabic calligraphy has long been considered an art form and people are judged by their handwriting.
Start an Arabic vocabulary book. Arabic has an extensive vocabulary and words have a lot of consonants that are difficult for non-native speakers to pronounce. Writing down new words and practicing using them is essential to improving your Arabic.
Watch Al Jazeera television. Coming out of Qatar, this television station is internationally known for its news coverage of current affairs. The presenters are chosen for their ability to speak modern standard Arabic. Unfortunately, they are always off-camera so you can't see the way they move their mouths when they pronounce the words.
Practice speaking Arabic with a Libyan accent as often as you can. If you can't find anyone to talk to, sit in front of a mirror and talk aloud to yourself. This may sound like a strange exercise, but it works. Walk around the house, point to things and say the name in Arabic.
Start an Arabic diary. Download an Arabic translation program and record your progress in Arabic. Get in the habit of writing in your Arabic diary every day to kick-start your language experience.
Read an online newspaper from Libya. Type "newspapers in Libya" into a search engine. There are a number of newspapers, so take your time and find the one that appeals to you the most and set it as the homepage on your computer.