In my opinion, Iraqi dialect is the probably the closest to classical Arabic(MSA). I also noticed that a lot of the grammar rules that exist in MSA are also respected in Iraqi dialect. That's not really a surprise given the fact the 2 of the most influential schools of Arabic grammar are both from Iraq (Al Basra and Al Kufa).
As Misslinda mentionned in her comment, Syrian dialect is also relatively close to MSA, but it is much less the case of Yemeni and Egyptian dialects.
Anyway, as both Omar and Taleb said, if you try to learn MSA first you should be able to easily learn most Arabic dialects.