Build a basic understanding of grammar and vocabulary. You can do this with almost any organized language learning course. See the Resources section for some possibilities.
Practice as much as possible. Your goal should be to link the Syrian Arabic words you learn not to their English translations but directly to the concept they represent. One way of practicing this is to put sticky notes with the appropriate Arabic word on everything in your house. So the sticky note with Arabic for "table" on it would go on your kitchen table and so on.
Enlist a native speaker, if possible, or at least another student, to let you practice back-and-forth dialog in Syrian Arabic.