I will often choose a different order depending on whether I'm trying a specific strategy and/or what the early events are.

But I'll often get L2 Engineering first, and L2 Science or L2 Explore as early as I can.

I'll hold off on L2 Farming until the food resources really need the extra boost of the coconuts, and L2 Medicine until my oldest villagers turn 50. L2 Culture always waits until I'm ready for L3 Culture (unless I get it in an event.)

The order of the L3 techs is often mixed up, and sometimes I'll have an L3 bought (usually Science) before some of the L2s.

IOW, there is no "correct" order, and changing what you do can make for some interesting games.