I got an alliance raptor on day 2, but it was quite painful. I started in Stormwind and from their ran south on a booty bay run. This is fairly easy till you hit the jungle. At this point you're probably gonna die, a lot, and you're gonna have a hella repair bill once you get to booty bay. I actually did this in order to obtain the fishing book. Once I was there I decided to explore some and took the ferry to Ratchet (a horde outpost). I was in luck and not many horde members had made it to ratchet on day 2, so I snuck into the barrens. This is where I first discovered raptors and I promptly abandoned my nightsaber to tame one. From here you can use your stone to get back, or hop back on the boat to booty bay and catch a griffin back to StormWind.
I'm now told that there is an easier way to get to the Barrens. This seems to envolved running through ashenvale to where it meets the northern barrens and then sneaking in and through to the lower barrens to catch a raptor.
Option 3 is to level to 25 or so and catch a raptor in the wetlands.
If you are on a PvP server I would highly recommend option 3 over the other 2, its less painful in the long run. :)