If you want to read Arabic, you must first learn the alphabet. Madinaharabic.com offers an excellent free introduction to the Arabic alphabet, as well as a free course in Arabic grammar. This course uses extensive examples with pictures to help you understand and internalize the structure of the language. The site also offers audio-visual flashcards and multiple choice quizzes to test your knowledge. You can even download the entire site at once or print off an e-book version if you want to continue studying while you are away from your Internet connection.
To understand spoken Arabic, you will need to practice listening to native speakers in various situations. In many areas, however, it can be difficult to find a native speaker to practice listening. To solve this problem, the University of Texas at Austin publishes a helpful program called "Aswaat Arabiyya" (Arabic Voices), which includes authentic samples from Arabic television programs. The selections are grouped by difficulty level so you can easily find samples that increase your knowledge without being totally incomprehensible. You can also change the videos' speed to help yourself understand them better.
Once you feel reasonably comfortable with the alphabet and basic grammar, it's time to practice reading. The British Broadcasting Company's (BBC's) Arabic service is a great way to do this because the stories generally focus on current news, so you can often guess a new word's meaning from context without having to search a dictionary. Because the they use straightforward, journalistic style, the BBC's news stories are also a good learning tool for students who are still mastering grammar and style in Arabic. In addition to the written news stories, you can listen to the day's news highlights or a webcast of what is currently on the air.
If you want to understand Arabic television, newspapers and official speeches, you need to learn Modern Standard Arabic (MSA), the modern descendant of Classical Arabic. However, if you want to speak with people on the street and in their homes, you will need to learn the local colloquial dialect of Arabic. For example, Moroccans speak Moroccan Arabic, which sounds different from MSA. Speakmoroccan.com offers a free course on this dialect and notes some of the major differences you can expect if you are transitioning from MSA.
Syrianarabic.com offers a course in Syrian Colloquial Arabic that includes audio files. Only the first three chapters of the course are free, but those chapters should be enough to help you decide whether the full course is worth your time. Gulfarabic.com offers a free course in the dialect of Arabic found in the United Arab Emirates, Qatar, Bahrain and nearby areas. The site requires you to register to access some of its content, but registration is free.