SF 340.
Left engine first for a GPU start because of reasons given by BH, and because the GPU is usually just behind the right engine, thus running the risk of frying the ramp rat as he unplugs it.
Right engine for a battery start, becase the batteries are on theRHS, so less voltage drop between battery and engine. Also, some aircraft have a prop brake on the right engine which allows it to be used as an APU for electrics and heating (as well as other services) on stand.