MSA Modern Standard Arabic, is the written Arabic used for all official documents. We learn it in school, we read and write in it. Classical Arabic is the same thing but the more eloquent, older version of the language, it was used to write poetry, religious texts.
As for which dialect you should learn Arabic in, learn it in a dialect that the rest of the Arab world understands which is either Egyptian or Levantine (Shami) Arabic because these are the two most popular and well understood dialects.
Khaliji, on the other side, is only understood by khaliji people, or by people close to those countries, not all of the Arab world.
Along with a spoken dialect of Arabic, though, you'd have to learn MSA if you want to understand newspapers, modern writings, etc.