There's a brief prologue explaining how humanity got to where they are, and their new world; a nice brief history that is no less accurate for apparently being part of one of River's hallucinations.
The movie starts off during Simon's rescue of River, and progresses to a point just as they are escaping, when we realize that we are actually watching a holgraphic replay of the escape, which is being viewed by a man referred to only as "the Operative". This (very dangerous, we come to learn) man was sent by the government to look into River's escape and to gauge the threat level.
As it turns out the threat level is pretty damn high, because the led scientist at the base/institute/whatever showed River off to some "high-level officials".
As the Operative says, "You put high level officials in a room with a psychic?"
He then kills the lead scientist in humbling fashion, and it is clear River is a threat to be dealt with.
We then cut to an indeterminate amount of time later, and Mal Reynolds is starting to use River on missions, much to Simon's displeasure. There's a nice moment where Mal has to get Jayne to leave the larger part of his arsenal behind on this mission, as the mission doesn;t really require grenades.
So of course, the mission goes south as Reavers appear. As Jayne says in the midst of a balls-out chase scene "Oh wouldn't it be nice to have some GRENADES right about now!"
They escape, but Simon is furious with Mal for the danger he put River in, and resolves to leave Serenity. They do so on the next stop - or would if things didn't go terribly wrong.
Mal is working on selling the proceeds from the previous mission. River is in the same dive bar where this is taking plac, when she catches a glimpse of a vid program - the vid has a subliminal trigger to it designed to set River off - and she goes "nutso-ninja-River" balletically kicking the ass of everyone in the place before Simon manages to shut her down (using a safe code given to him by one of the people who helped him break River out) just before she was about to put a bullet in Mal (IIRC)...