Derailleur adjustment: front or rear first? When adjusting both front and rear derailleurs, which should be done first?
If the derailers just need minor tweaking (they are basically in adjustment but are "not quite right") then you adjust the one that is obviously wrong, check the overall adjustment again, then again adjust what needs adjusting.  Often only one derailer needs adjusting and often the adjustment is just a half-turn of the barrel adjuster.

If they are completely mucked up (but nothing visibly bent/broken), I usually first check the front for parallel and sprocket clearance (they often get knocked out of alignment), get it on the center sprocket, then address the rear.  Once the rear is basically correct, get the front to where it will shift to all rings with modest accuracy, then, trying all combos, adjust the rear some more.  Finally, fine-tune the front, then fine-tune the rear.

But it varies with the bike, how recalcitrant the derailers are, and specifically what is out of adjustment most.