My starts are usually:
(1) Have Mining already: Worker first off while learning Bronze Working. Chop a settler, preferably while actually working a warrior or work boat and switching to the settler when the tree hammers come in.
(2) Don't have Mining yet: Learn Mining first. Make a worker if there are any tiles that I can work with my starting techs (Rice/Corn/Wheat and I have Agriculture, or Deer/Elephants/whatever and I have Hunting), but more likely make a Warrior and grow to size 2, and start the worker such that he pops right around the turn where I learn Bronze Working. See above for popping out the first settler, though weaving the hammers with a warrior is less important if I already have one to escort the settler through the animals.
Main time I'll deviate are if I'm going for religion or I have a UU that requires Horse- in which case I want to know Animal Husbandry before I settle that first settlement anyway, so I might as well grow a bit first.
Get the capital up to about size 4, then churn out 2 more workers and then another settler. Somewhere in there we whip granaries in both cities.
I had always timed my worker to pop out so he can chop my settler, and I usually don't switch to Slavery until the settler's out because I don't tend to have necessarily grown yet (unless I'm spiritual). But this thread has be reconsidering my build order so I can get to size 2 before I start the settler and then whip the settler in addition to chopping it.
Edit: I usually play multiplayer, and one of our other players plays on Warlord, so the AIs tend to be around Noble or so. In single player games I've recently moved up to Monarch, so take the above with the appropriately sized grains of salt if contradicted by an Immortal player.