Start with Civilization Revolution on the XBOX 360. Give it a rent and see if you like it. If it seems tedious or unentertaining, you'll absolutely despise the computer versions of Civ. If you don't own an XBOX, start with Civ4. But before you do, I'd recommend visiting www.civfanatics.com
and reading their strategy guides. They have a million of them on every possible subject, as well as a great beginners guide. Hope this helps.
Rev is also available on PS3 and DS, the DS one's supposed to be just as good as the console versions as well.
EDIT: Wanted to add that I think Rev is also the most "Noob friendly", and has the most personality.