I've been levelling an alt recently, and found that the /general channels of level appropriate zones have been the best place to start.
There's often enough other people levelling there that some would be interested. Once you've got 3 or more you'll know whether you're still short of tanks, healers or dps, and can start using /who (e.g. /who priest 35-40) to identify people in other zones that might be capable of filling your empty spots.
If you do take to whispering strangers, make sure to be friendly and polite, and don't spam them more than once. Something like "Hello, would you like to join a group to Uldaman? We're in need of a healer" would normally get friendly answers.
You should also post a LFM in the LookingForGroup channel, as many people levelling alts are inveterate LFG-lurkers and may well be interested.
If you're willing to take the initiative and form the group, then it shouldn't take too long if you do the above (unless you're on a very low-population server, or its 5am or something).
Instances while levelling are a great way to populate your friends list, and the bigger the list, the easier it is to get a group the next time.