SCII lore was really terrible & simple to follow, despite being the first part of the trilogy. You don't need novels to follow, they're just supplemental.
Let me cliff it for you:
Before SC1, in the CD sleeve game manual:
Earth became united as one nation, they send half a million prisoners to new planets. They crash land, and settle there over 3 planets, known as Terran.
Xel'Naga is a godly master race whose hobby is to create a perfect race. They create Protoss first. They watch Protoss race prosper over the course of many millennia as they coached them. Later on Protoss got headstrong and rebelled against them. A big war happened and Xel'Naga disappeared.
Xel'Naga tries again- an adaptive race called Zerg. Overmind is made too. But they also rebelled. Xel'naga disappears. Zerg grows bigger by devouring planets.
Events of SC1 original:
Terran is having a civil war, The main entity called Confederacy is too oppressive, and other two factions in two planets are pissed off too. Suddenly Zerg arrives in their sector and rapes everything. They also witness Protoss ship destroying Zerg too. They're pretty much WTF. A backwater marshall named Raynor join forces with the upstart led by Mengsk. They are also joined by Kerrigan. They overthrow successfully and Mengsk is crowned an emperor. But, Mengsk proved to be the same shitty dictator same as Confederacy. And he also sacrificed Kerrigan to Zerg. Raynor escapes.
Zerg got a hold of Kerrigan and turned her into a Zerg hybrid creature. She became the right-hand person of Overmind much to the disapproval of other Cerebrates. They basically wreck everything including Protoss' homeland Aiur, and settle there.
Protoss is having a civil war themselves due to their proud nature. A high templar named Tassadar seeks out Dark Templars who have the power to destroy the Cerebrates and Overmind. Protoss council is mad at him for what they feel is betrayal. But in the end they learn. Tassadar also met Raynor in the midst of a Zerg battle and they join forces.
In the end, Protoss and Terran join forces to destroy Overmind. Tassadar sacrifices himself in doing so.
SC1 Brood War:
Protoss struggles to reel back from Zerg's destruction of their homeland. With Overmind dead, they fight out of control Zerg forces. Kerrigan approaches Protoss as says Overmind is gonna come back. She eventually earns Protoss' trust through doing side missions (retrieving Psi disruptor which controls Kerrigan & helping kill Zerg).
Of course Kerrigan was using them. With Psi Disruptor gone and Overmind's death assured, she assumes total control of Zerg. She totally freaks out and destroys everything & kills off few important side characters too (Phoenix & Edmond Duke).
Raynor is seething with rage and vows to kill Kerrigan.
SC2 Wing of Liberty:
It's been 4 years. Raynor is a bitter worn man. Zerg has been quiet for an unknown reason. Mengsk has marginalized him as a #1 public enemy through propaganda. Zeratul appears for the first time since Brood War and says Xel'Naga is coming back to rip shit up. The master race will be the undoing of all of us, nevermind Zerg.
Most of the missions are terribly insignificant to the story. Raynor collects various ancient artifacts as Zeratul instructed, fends off random Zerg infestation, and puts the artifact together to experience a vision which shows Xel'Naga fucking shit up. Then Zeratul says you can't kill Kerrigan, she's the only key to stopping all this.
Kerrigan & Zerg are back, with no explanation as to why she didn't finish everyone off 4 years ago (what a shitty writing). Raynor is approached by Mengsk's son who doesn't like his own father. They take half of Mengsk's total army and launch a full-assault on Zerg's original homeland Char.
They defeat Zerg with the artifact and Raynor 'rescues' Kerrigan who seems to be semi-reverted to her human state. The End. What a subpar lazy-ass writing compared to all the epic stuff happened in SC1.
SC2 Heart of the Swarm & Legacy of the Void
It's been revealed HotS will spend majority of the time with Kerrigan trying to regain control of her human self and reclaming Zerg (wtf shitty writing and didn't we do this already in SC1?). I suspect it will end with her killing Emperor Mengsk, who betrayed her and turned her into this abomination she is today.
Here's the leaked ending:
http://www.youtube.com/watch?v=I14VFbdSoCI
Legacy of the Void will obviously focus on Protoss and all races coming together to destroy the common enemy Xel'Naga. The same rehashed formula Blizzard has been using since Warcraft 3 (Human, Night Elves, & Orc come together to kill Undead & Burning Legion).
Answering your questions:
In the OP's trailer, Kerrigan was obviously dreaming or recalling the past. It's a generic battle, and based on the human buildings and the Mengsk statue, it's probably Tarsonis. She wakes up to find herself as 'human' again. So this is immediately after Raynore rescued her at the end of WoL. She is recovering in Raynor's battlecruiser.