I got my first pet the day I started WoW.
Here's the abbreviated story:
I came to WoW late January or early February after release to join some friends from EQ. Once I got past the newbie area, and got to a mailbox, I collected the (AMAZING!) 5g that I was sent by one of the friends. Then the friend's wife showed me around IF, and took me on the train (while demonstrating the adorable /train command). In showing me around, I started fishing and cooking. Then she took me over to the cat lady, where I bought my first pet (the Bombay), and that was just too great!
I didn't have enough bag space to buy them all, so I contented myself with the one to start.
Soon after that I saw the panda cub, and was disappointed to learn that I could only get that cutie by having the collector's edition and I was devastated. However, I'd already been spending a lot of time on Allakhazam's site looking things up (to get info on how I could get a kitty mount, but that's a different story), so I looked up pets too. I ran across mention of the crimson whelpling, and started my quest for that. I slaughtered them by the hundreds from level 23 to 30 until I finally got my chubby-cute baby dragon. Then I thought to myself... I like green better than red, so I went and started camping the emerald whelpling. Which carried me to level 41, where I thought, I might as well get the dark one, and got that one 4 kills after getting the emerald. I leveled, and knowing I had limited space, avoided the storebought pets (mostly), and collected what I could of the rest of the pets. Any time we wiped in a raid, I would change up the color whelpling I had out (since obviously it was the wrong color for the encounter).
When I started my hunter alt I didn't want to be without a whelpling, so I camped the crimson (to match her hair), but decided I really did like green best, and then since I had two, I may as well get the third... This carried over to every alt I made.
By the time vanilla ended, I had 10 alliance toons on 2 accounts, each with all 3 whelplings, plus an assortment of other pets (each toon had 1 pack devoted to just pets for everyday, plus bunches more in the bank), I also had a set of whelplings waiting for the new draenei I was going to make when BC was released.
The announcement of the pet tab at the end of BC sent me into a frenzy, I went around and made sure I had all the vendor pets, plus as many other pets as I could make or otherwise lay my hands on for each and every toon on both accounts, at which point I created a spreadsheet to keep track of which toons had which pets and so on.
At some point, I did purchase the WoW laptop (approximately 1 or 2 days after they'd been discontinued), so I got my original CE pets that way, and gave the BC CE code to a friend.
At the moment, I have 2112 pets spread across my 18 toons (that's an average of over 117 per toon).
However, due to a variety of factors combined with the announcement of pets going account wide for MoP, I'm reigning in my enthusiasm (with difficulty) and just making sure I have as many pets as possible on each of the WoW accounts.