Why is engine No.2 started first, instead of No.1?
The isolation valve in Open or Auto ensures that the Auxiliary Power Unit (APU) bleed air reaches the engine number 2 starter. Engine number 2 start Exhaust Gas Temperatures (EGT) are generally higher.

http://www.lt-vacc.org/images/mesti/Boeing%20737-300-400-500%20Operations%20Manual.pdf

Engine number 2 is started first because it's on the opposite side of the air bridge. When starting with external air or power it is normal to start Engine number 1 because it is further away from the carts.

http://theflyingengineer.com/projects/airbus-tech/hydraulic-systems/

According to Flight Crew Operations Manual (FCOM) 3 Standard Operating Procedures (SOP): 


  Engine 2 is normally started first. It powers the yellow hydraulic system, which pressurizes the parking brake


Looking at the electrical schematic that Boeing provides, we see that Transformer-Rectifier (TR) 3 is powered from the R/H AC Bus. TR 3 is an important TR from which significant avionics are powered. Ergo, you would want to have this TR powered up and running first in case you lose AC power to the aircraft during start.