I seem to like version 4 or 5 the most of the Civilization games. I have not yet got into CIV VI but it seems most people playing today still play 5 for multiplayer.
I love the multiplayer version as it is the only version I have played. I have played CIV since version 2 or 3 but version one is a very classic game like pac-man for atari as far as online gaming goes.