Originally Posted by Lamont Cranston

It became self aware @ 2:14 a.m. Eastern time, August 29th, 1997.
Presumably some time later in response to the remaining humans it (somehow) used surviving automated factories to produce war material, although Terminators and HKs wouldn't have been the first things off the
assembly line. Probably construction robots.

Fun facts: on the commentary track, Cameron says Skynet set up the whole time travel thing to ensure it wouldn't be created - it regretted its actions. And in the first draft of the Terminator 2 script, the Terminator has a vision of walking towards a bright light after jumping off the gantry into the molten steel.