What's the story of 19-2 (either version)?
The story is the sum of the individual stories of every cop in the 19-2 precinct so there is not one unifying thread until the end of season 1.
The great thing about the show is that it doesn't tax your suspension of disbelief. If they turn on a perp's computer it's not going to be a fancy hollywood UI on it like in CSI or the canadian 19-2, it'll be plain Windows. Just like what you expect.
Lots of research was put into the show and cops find it very believable.
Season 2, episode 1 was based on the question "What is the worst thing that can happen to a cop?" and modeled on real events.