Actually, one should NEVER move their initial settler so they can't settle on their first turn,.. I mean, you lose SO much because of that.
Beating a dead horse by this point, but I had one game wherein my starting point was within two tiles of a mountain, next to the ocean with numerous hills to its left and wide open space with a river to the right. If I had settled then and there, I would have had access to Machu Pichu, two luxuries and navy access. But, and this is important, there is a hill directly beside my starting spot next to the mountain, with an extra luxury within the third ring.
So to summarize: I can settle where I am, with access to two luxuries, Machu Pichu, Neuschwanstein, and a navy, or I can spend one turn to settle on a hill, next to a mountain, getting me access to three luxuries, an Observatory, Machu Pichu, Neuschawanstein and a navy, with an added +1 production from settling on a hill.
You lose five or so science, three to five gold, two to three food and one culture in moving your initial Settler, multiplied by every turn you spend moving (though with increasing opportunity costs). Simply settling on a hill, however, provides incredible advantages that overpower the losses (You gain your initial Monument/Library/Worker/Granary nearly
twice as fast, you do not have to work the hill tile as it will effectively function as a farmed hill tile upon settling, and you have increased city defense, to enumerate the obvious bonuses).
In the above example, the option was made even more obvious by the fact that I had a mountain right next to that hill. A Tall, Observatory-ed capital is well worth a few turns spent looking for that spot. Add in the fact that I had more luxuries had I spent one turn moving, and you can see that the option of moving was a no-brainer.
The early game is important, true. But the early game yields are, truthfully, miniscule , with, however, exponential potential for growth. The option of moving, as is the option of settling where you stand, is always situational, with the right option being based on the situation and condition of the current environment.