The premise is their existence and that they threaten Earth because there are critters out there who are powerful and use humans as hosts. So what do people do about that?
^ Travel through the galaxy putting up wanted posters for Dick Cheney?
There were several layers to Stargate, the Ancients who built the network (for reasons never entirely clear), their contemporaries who mostly disappeared (the Nox, Asguard, and the never-used Furlings), and the parasite Go'uld who came later. Then they added the Ori and the Wraith.
The Furlings are still an unexplored element, and I once roughed out an idea for them where the SG-1 team travels back in time to discover that Furling agricultural scientists were modifying a primitive parasite (the Go'uld) so they could control farm animals (space cows!) so the animals wouldn't need Furling supervision in harsh farming environments and would be immune to disease and resistant to injury. But the ag scientists didn't want to have to train each new symbiant, so they added genetic memory. To guard against criminal use or out-of-control parasites, the scientists made sure that Go'uld couldn't survive very long in a Furling host. Then there was a lab accident... The Go'uld quickly took over important Furlings and destroyed their society with Furling weapons, and then had to find compatible hosts. Most of the lab-enhanced Go'uld died, which was why their race was dying when they first encountered Earth.
But that would get you a stand-alone movie, not a series.