Which barrel of a side by side shotgun should fire first?

Assuming your shotgun has two triggers, you can fire either first.

If you have one trigger, the order of fire can vary massively...use a gauge to check the bore on your gun, If you don't have one just use a coin and see how far it will go down the front of the barrel, or make a target out of cardboard to see which barrel is the more open barrel, by patterning the barrels.

once you know which barrel is the the more open choke, you are ready to determine which barrel you want to shoot first, and as a quail hunter on the rise you will usually shoot the barrel with the more open choke first, saving the tighter choked barrel , as the game gets farther away.

Source: shotgunworld.com