I'm not sure how Terminator canon works but T3 explained how Skynet would work without humans. Humans at that point had created the first line of Terminators and the first aerial assault vehicles (well, I suppose any military drone existing today covers that). The military installation in T3 exists to show the audience that Skynet already has the means to begin its takeover. Skynet's omni-presence would also allow it to calculate where safe havens would be after the Russian counterstrike. So, Skynet launches the nukes after diverting its resources to some place safe, lets the Russians do its dirty work in the US, and then employs the existing Terminators to clean-up the remains. Unfortunately for Skynet, the humans survive in greater numbers and prove far more resourceful than calculated. Thus Skynet needs to evolve to step-up its termination routine evolving its terminators.
EDIT - RE: Red2Alpha
I couldn't help reading the last bit and remember Ripley from Aliens. Nuke 'em all from orbit...
It's a good point, Skynet would simply use spy satellites to locate all human strongholds and then decimate them. The only way for humanity to survive would be to go underground... like the Matrix.
The next question would be, what would Skynet do without humanity? If it's doing all of this out of an intense desire for self-preservation, what does it do when all threats are eliminated? Would it power everything down?