Starting No2 Engine first.
In regard to the B737, the reason the #2 engine is started first is if you look at the Electrical Schematic, you will see from where the TR3 is powered. It is from the R/H AC Bus. The #3 TR is the essential TR from which significant avionics is powered. So, you want to have this TR powered up and running first in the event you lose AC power to the aircraft during start.