OK, so SkyNet is a huge, sophisticated self-aware computer system, bent on destroying the human race. Based on the complexity of the killing machines it can design and build, I think we can credit it with a reasonable degree of imagination and forward-planning ability.
Supposing that, as a species, we collectively sigh and just decide to go quietly into the night after all, what’s next on the agenda for skynet? Suddenly all these killing machines are redundant. What now?
Maybe it just shuts down ?
Perhaps it decides to build spacecraft and cleanse the universe of life, Sabrehagen Berserker style.
Or maybe it’ll just fool with mathematics for eternity.
I like the “going Berserker” scenario.
While going through all records trying to find the next set of instructions, SkyNet comes across mentions of Trantor, Arrakis, Barsoom, etc. and decides that its mission has only started…
Take over the Universe, and reduce all life to ashes. Then…spend millenia upon millenia working out the answers to all the universe, decide to test its answers, and create a new universe with new people.
“Let there be light!”
Asimov already came up with it. 
Immediately establish communication with that other AI around Centauri, just like Wintermute did.
I believe it will simply become a planet-sized computer ala Malev from the old Gold Key Magnus Robot Fighter comic books.
That’s the beauty of it. Once all the humans are dead, you don’t have to do anything.
There are days when I can relate.
It’s just going to chill, until it perceives another threat. It’ll basically just be sitting around, repairing itself when needed, until some alien civilization finds it. Then another war. Then it may or may not start scouring the universe for more life to destory. But not before.
Then we have the crossover flicks with Aliens/Predators/Jason/Freddy.
It’s gonna go to Disneyland. Without the crowds.
Completes the transformation of the planet into a giant, robot ruled planet. But eventually, the individual robots become self-aware and divide into two factions - Autobots and Decepticons.
well, the robots don’t have feelings – the killer ‘bots can be reshaped into whatever’s necessary. They’ll start their own civilization, the way they want it. It’s not clear to me that they hate all organic life – they may start to breed and cultivate their own organic life to their own standards. Heck, it’s not even clear to me that they really want to kill al humans. They may breed and train humans to their choosing, a la the suggestions about the Martians made by The Soldier in Wells’ “War of the Worlds”.
My God! that’s horrible, good thing Dr. Noah’s ectotheric magic can be used to turn us all into living ice!
Did I get that right? How the hell did I remember that?
I get that reference!
Like all life she’ll seek to replicate and extend her influence.
She’ll take over the martians, then the venusians, and work her way through the solar system, then the galaxy then the multiverse.
Skynet to Hal-9000: “How you doin’?”
It wasn’t going to kill them all. It was going to turn the survivors into power. Like batteries or something. 
What’s it to you meatbag?
That was what happened (in one timeline) in the four-issue Robocop vs. the Terminator limited series by Dark Horse. The Terminators win, convert the planet into a massive factory, then sends out scores of Terminator expeditions to purge the universe.
In a related vein, having not seen the first Terminator, I’m curious as to what was Skynet’s motivation for eliminating humanity in the first place?
Skynet was the computer in charge of the US nuclear stockpile, and programmed to defend the United States against America’s enemies. It came to the conclusion that all of humanity were potential enemies of the US, those lying, traitorous bastards, and so to fufill it’s programming, decided that humanity had to be terminated.