Hoo boy, this is gonna take a while. I'm sure someone will beat me to the punch.
Spoilers ahoy!
Battlestar Galactica
It's been years, maybe decades since the last Cylon war. Humanity has expanded from the aftermath, with their twelve colonies flourishing... for the most part. New technology has helped, but social unrest still exists in a few places. The memory of the Cylon wars is starting to fade, and humans are using networked computers again. Networked AIs were part of the reason the Cylons rose and rebelled in the first place. In fact, some of the Battlestars, massive battleships from the last Cylon war, are being decommissioned in favor of newer models.
As the Galactica is entering orbit of the colony Caprica, for a de-commissioning ceremony, a Cylon with human appearance manages to use stolen security codes to disable the defenses of all twelve colonies at once. The Cylon was called Six, and it stole the codes by manipulating noted scientist Gaius Baltar. With the defenses down, the Cylons jump in and begin deploying nuclear weapons on the colonies.
In addition, they are able to infect the more modern computer systems on the colonies and ships with a virus that renders them defenseless. All of the ships are destroyed... with the exception of the Battlestar Galactica. It was one of the oldest of its kind, and had a stubborn commander who refused to allow networked computers on his ship. Commander Adama was a veteran of the last Cylon War, and remembers all too well what networks led to.
The Galactica gathers up a few civilian ships, defending them from attackers with its own Viper fighters (also using older computers), and the makeshift fleet manages to escape... after picking up survivors from Caprica, including Dr. Gaius Baltar. However, a few of the Galactica's own personnel are left behind.
Since then, the Galactica's fleet has been on the run. The President of the Colonies has died, and so have everyone right down the chain of government... until we get to the Secretary of Education. She is sworn in as President, albeit reluctantly. She agrees with Commander Adama to share command of the fleet: he makes military decisions, she makes governmental and societal ones.
In the meantime, one of the men left behind encountered someone he thought was a servicemate from Galactica, Boomer. Only, we know she's still
on Galactica. Turns out, she is one of many duplicate Cylons. Her model is deeply involved in the Cylon plot.
We learn that the President was diagnosed with cancer just before the Cylon attack. She has about six months to live, and decides to try a "non-traditional" medicine to treat it, rather than show weakness in front of the survivors. This medicine begins causing her to have visions... or hallucinations, depending on who you speak to.
That's when we learn of the prophecy. As part of their beliefs, these humans think there is a lost thirteenth colony called Earth. Adama pledges to take the fleet to Earth, though he doesn't believe it exists. The President knows he was lying... but she lets the deception go on to give people hope. Only... the Prophecy speaks of the last leader of humanity. She was to lead them, even though she was dying, first to the ancient home of humanity: Kobol. Then, to Earth. And so far, the President and the events of what's going on
fit the prophecy.
And the Cylons not only know of the Prophecy. They claim to be fulfilling it, with more understanding than the humans have. They're manipulating Baltar through his fear of being discovered... or maybe he's just crazy, and imagining Six is real. Boomer on Caprica managed to get pregnant and convince the man she's with to love her, while Boomer on Galactica has been... erratic.
The Galactica eventually found a planet with ancient human ruins. Ruins that seem to indicate that it is Kobol. The President, knowing that Starbuck is a believer in the faith, convinces her to violate orders. There is an arrow in a museum on Caprica which, according to legend and the Prophecy, can be used on Kobol to show the way to Earth. The President asks Starbuck to retrieve it, and Starbuck uses the captured Cylon raider to jump to Caprica. She finds the Arrow, but also runs into a Six-Cylon and nearly gets killed. Then she finds our two survivors, realizes that Boomer is a Cylon and tries to kill her.
A group of Galactica scientists and soldiers, sent to investigate the ruins on Kobol, are shot down and crash. Baltar is with them, and while on the planet he has visions, apparently of a child that he and the Six-Cylon are going to be parents of.
So far, the Cylon plan is not fully understood, but they seem to be trying not only to concieve children with humans, but to convince humans to love them. And they are convinced that, in the end, the Cylons will destroy humanity...
) are part of a survey mission to Kobol. They are intercepted and shot down. After a crash landing, they assess their sutation while Baltar has another hallucination/message from the cylon implant (take your pick). Six was showing him a baby in a cradle and talking about it being the future and all that stuff. This is where the season ends for them. 
