Arabic is what linguists call a diglossic language which means that the language spoken in daily life differs from the written version and varies widely from country to country.
In modern everyday life in the Arab countries, Arabic dialects have replaced Standard Arabic for spoken communication, but they are all derived from Standard Arabic, so if you have a grounding in Modern Standard Arabic (MSA), it will be easier to learn the modern local dialects that are based on it.
MSA is also the lingua franca of the Arab world, and can be understood anywhere in conversation with educated speakers.
Where to start from?
There are a total of 12 parts in the sciences of Arabic. The 3 most important parts out of the 12 are:
1) Sarf ( morphology )
2) Nahw ( grammar )
3) Balaghah ( rethoric )
Generally Arabic teachers, language centres or guide books focus on Nahw and Sarf because without it as basics it is difficult to understand Balaghah.
Nahw and Sarf always come hand in hand.
The common guide books:
Madinah Arabic Course Books by Dr. V Abdur Rahim
Al Arabiyatu Bayna Yadayk
Al Kitab Al Asasi