What we know for sure about the Doctor's origins -- and I'm going to go ahead and put this in chronological order for his timeline, though of course it's instructive to look at the order in which these facts were revealed on the show, too:
He's a Time Lord from Gallifrey. He has two hearts, and when he dies he regenerates into a new form. He's currently in his eleventh incarnation of thirteen.
He and the Master went to school together. When they were children they looked into the Total Perspective Vortex; the Master went mad and the Doctor ran away.
The Doctor and the Master chose their own names.
The Doctor stole the TARDIS and fled Gallifrey. Until he finally gives himself up in The War Games, he is a wanted man, both for the theft and for violating the Time Lord equivalent of Star Trek's Prime Directive and constantly interfering in affairs of the universe instead of just standing by and watching.
When we first meet the Doctor, he's parked his TARDIS in a scrapyard in 1963 London. He's an old man traveling with an apparently teenage girl who goes by the name of Susan Foreman and calls him Grandfather. (We reasonably assume that he is in fact her biological grandfather, though this is never stated unambiguously.)
The reason he's in London in 1963? He's got a powerful Time Lord artifact called the Hand of Omega -- a device which is capable of destroying planets across time and space -- and seeks to hide it.
I don't know if that's an exhaustive list, but those are the key points of what we know about the story before the beginning of the first episode, off the top of my head.
Other than that, there are hints, and things expressly stated as the intentions of the writers (and, in some cases, novelized), but nothing really concrete on the show itself.
First, there's the nature of the Hand of Omega. It's the device that Omega used to perfect time travel by collapsing a star for fuel. Omega was blasted into the antimatter universe, and his partner Rassilon became the first Time Lord.
The Doctor later says of the Hand of Omega, "And didn't we have trouble with the prototype?" Pressed by Ace, he corrects himself and says "They." Script editor Andrew Cartmel has stated explicitly that his intent was to imply that the Doctor himself was a third founding Time Lord, alongside Rassilon and Omega (and there are further seeds to this end in Silver Nemesis, which I've never actually seen), but this plot never reached fruition.
Not on the show, anyway. There's a novel called Lungbarrow which lays the whole thing out; I haven't read it but it sounds immensely complicated, implying that the Doctor is a reincarnation of the Other, retaining his memories through the Time Lords' technological method of asexual reproduction, the Looms. It all sounds frankly quite complicated and is absolutely not the sort of thing I would ever expect to see on the TV show.
But the idea of the Doctor as the Other? Still a possibility. Especially since I expect we'll be seeing Omega again soon.
The Omega symbol's been cropping up on the show for several years now. The clerics who kept River as their prisoner in Time of the Angels had it on their uniforms. It cropped up again in A Good Man Goes to War.
Now, obviously that shit's there for a reason. It's not just there because some costume and set designers think omegas look cool. And while certainly there's a religious connection, this is Doctor Who; there's no way they threw it in just as a Revelation reference.
So I'm comfortable in assuming Omega is going to figure into the coming story arc somewhere. If Omega himself doesn't put in an appearance, then his legacy's going to be relevant, and I bet we'll be hearing about the Hand of Omega again.
Last night I ran across the possibility that the Doctor's name was a mathematical formula. What if the secret of time travel itself is encoded in his name?
Again, that would make it tough for him to be one of the three founding Time Lords (short of a paradox where he travels back in time and becomes one of them) -- but I can easily imagine that Time Lords might name their children after founding principles of their society. Like naming a child after a religious figure or a king or a lord.
It would also explain why the Doctor had to change his name before going out into the wide universe -- if his name holds the secret to time travel, well, that wouldn't be a secret among the Time Lords, it would be common knowledge. But sharing it with somebody outside Time Lord society would be a big no-no.
We've seen, of course, over the series, that there are other races and cultures capable of time travel -- but theirs is always rudimentary, limited in some fashion. Dangerous and unpredictable (moreso than the Doctor's TARDIS). The Doctor, as last of the Time Lords, guards his TARDIS and its secrets closely -- and maybe that includes keeping his name a secret.
It's an elegant theory and I like it -- but it creates its own problem: barring a copout where his name is never actually spoken onscreen, he still needs to have a name. And, if this theory is correct, it needs to be a name that's not just satisfying to the ear but also sounds like plausible enough technobabble that the audience will buy that it's some fundamental secret of time travel. And you've only got a short space to work with. (My full name is nine syllables. And it's pretty long.)
Alternately, maybe it's something far more prosaic. Or maybe Moffat's got something much more interesting in mind. Guess we'll find out in three weeks.