I logged in for my once-yearly Civ Fanatics post, because this topic is so delightful
I started with Civ 1. I still have the box for the PC edition from Microprose, though I can't find that giant manual. The first time I played it, I started about 9am, and the next thing I remember hearing from my family was "dinner's ready!" I believe it was summer 1993; Civ 1 and Star Control 2 were the PC games that ruled my summer that year. Key memories include those blood red barbarians, refusing to build churches because I was deep in my rebellious anti-religion teenage phase, and Teddy Roosevelt advising me when the "Babylonians use nuclear weapons!!"
I bought CivNet and played it with a friend once or twice, but mostly it was to have a higher-res version of Civ 1.
Civ 2 was the first game I ever pre-ordered over the internet, from Egghead Software as I recall. I still have that box, too; I'd just returned from a trip to meet some gaming buddies from IRC in Chicago, and the Fedex package was waiting for me when I got home. It was a great way to return
Civ 3 was the game that broke in my Athlon XP 1600 in October 2001. Unfortunately that box is lost to the sands of time.
I took a day off from work for Civ 4, but was wrong by a day

So I only got half a day or so to play it instead of starting the night before

The boxes for Civ 4, Warlords, and BTS bookend my collection of gaming relics at the moment.
I never played any of the CtP games. I played Alpha Centauri, and while I appreciated it for what it was, the flavor of Civ has always been important to me, so it didn't grip me quite the same way.
Civ as a series is one of the few games that I can say has improved with every iteration. That makes me very optimistic for Civ 5. Yes, there's a ton of changes coming, but the changes over the years have made the game more fun. Civ 4 in particular, with its emphasis on eliminating unfun elements (pollution whack-a-mole, cities in disorder) really raised the bar and remains my favorite iteration of the series.
For reference, since age is coming up a lot in this topic, I will be 35 next month.