The information about you flipping the switch will have to propagate so that current can flow. Thus the bulb the closest to the switch will light up first.
Think of it in the water analogy. You have a long channel, with a gate in the middle. One side of the gate is flooded (high voltage), the other is dry (low voltage). If you flip the switch (open the gate), where will the current flow first? Of course the water the closest to the gate will start to flow first.