Many western scholars distinguish two common Fus′ha varieties: Classical Arabic, that was used during the 6th and 7th centuries A.D., and Modern Standard Arabic, the standard written language used today (not the vernaculars, though).
The word Fus'ha is used to indicate non-vernacular or correct Arabic.
Which level (fus'ha or a vernacular) is spoken often depends on formality.
As people say, you will get strange looks speaking in fusha as people do not use it as a spoken language on the street, but it will be a help until you can pick up useful phrases.
I was advised that it is best to learn fusha first and then a dialect, as it is more complex and therefore makes learning the dialect easier, but it depends what your intention is for learning it, whether to speak to people or to read and write.
I heard someone speaking once in what was supposed to be Arabic but because it was fusha and the 3ein's, 7a2's, and hamzas were all gone I first thought it was greek or something.