Find an example of the Arabic alphabet. You can use an Arabic grammar guide or find an online source. Websites such as Omniglot.com have available materials that show how the Arabic alphabet corresponds with the English one.
Practice writing the Arabic alphabet. Unlike most alphabets that have only a single form for each letter, there can be up to four different way of writing a single Arabic letter. As you become comfortable writing the individual letters, move to writing words.
Become familiar with speaking the language. While nearly all of the letters have an English equivalent, the sounds they produce are different. Don't start with complex sentences. Instead, begin with learning to properly pronounce the alphabet, then move on to simple words before tackling more complex ones. Be aware of what you are saying rather than simply mimicking sounds.
Learn vocabulary. Start with the easier words (girl, boy, tree, dog) before moving on to more difficult ones. Separate the vocabulary into topics (foods, colors, clothing) so that you're learning a group of words rather than single, random words. Test yourself frequently.
Study the grammar. The formation of the sentences will differ depending on whether the sentence is spoken or written. Learn the nouns, personal pronouns and tenses and how they're employed in an Arabic sentence.
Try composing your own simple sentences in Arabic. Use all of the grammar you've learned as a guide to help you with your composition. As you become more comfortable with it and you're making fewer and fewer mistakes, try more complex sentences. Convert your written sentences into spoken ones (remembering to alter the grammar slightly) and say them aloud.
Find someone who speaks the form of Arabic you're studying and try having a conversation. Community and cultural centers are an excellent place to find people who are willing to help. A native Arabic speaker will also be able to help you learn less formal versions of the language (slang) so that you sound more like a native speaker.
Read Arabic books and watch Arabic programming. The more that you familiarize yourself with the language, the easier it will become to speak and write it. Speak the language whenever you get the chance.
Branch out into studying other Arabic dialects. Since there are some similarities among the dialects, use your familiarity with your chosen dialect to help you. The more you study, the more advanced your knowledge of Arabic will become.