From my own personal perusal of the series since... well, a long, long time ago... it seems the following is pretty much canon:
- The Doctor is a Time Lord
- Time Lords of Gallifrey can only regenerate 12 times, meaning there can never be more than 13 incarnations of a Time Lord. (source: "The Deadly Assassin")
- The so-called "Rule of 12" has been broken before by Rassilon (the revered demi-god like founder of Time Lord society) and the Master. (source: "The Five Doctors" and the later return of The Master)
- It's not uncommon for some Time Lords to adopt a pseudonym in the form of some kind of title ("The Doctor", "The Master", "The Rani", "The Corsair")
- There have been 11 incarnations of a Time Lord calling himself "The Doctor" (source: various, mainly BBC statements)
- The Doctor once had a family including parents, siblings, a wife, children, and at least one grand-child. (source: mostly conjecture from statements made by Russel T. Davies regarding "The Woman" in "End of Time" pt. 1 and 2. The only confirmed family was his grand-daughter Susan from the 1st Doctor's era)
- All of the Doctor's family was dead (or believed by him to be dead) by the time of the Ninth Doctor. (source: multiple references made by the 9th and 10th Doctors)
- The Doctor talks a lot about laws of time but breaks them a fair amount. This includes crossing his own timeline more than once. (source: "The Three Doctors", "The Five Doctors", the "Trial of a Time Lord" arc, "The Two Doctors", etc.)
- At some point, a being will come about that is a distillation of the Doctor's "dark side". This being, known as "The Valeyard" will manifest between his 12th and final incarnations. (source: "Trial of a Time Lord" from the 7th Doctor's era)
- The Time Lords are all gone, as far as the Doctor knows. They are locked in the Time War.
- There are quasi-Time Lords out there. Namely Jenny ("The Doctor's Daughter"), Donna Noble (has some kind of Doctor-Donna meshing thing going on there), and River Song.
- The Valeyard somehow manages to travel back in time to Gallifrey from at least five incarnations in his future to cross his own timeline and try to steal the life of the 7th Doctor (probably due to his fashion crimes).
- Spoiler: There is another incarnation of the Time Lord who calls himself "the Doctor". This incarnation committed some kind of horrible crimes and is not considered "The Doctor" by his other incarnations.
So what if the Doctor was fleeing Gallifrey to escape some kind of persecution, perhaps for a crime of some sort? What if the Doctor's family was executed or killed as part of this crime? So you have the Doctor fleeing with the last remaining member of his family and getting the hell out while he can.
Oh, it's not canon, but it gives a more feasible explanation for an elderly being committing a crime punishable by death in his society and dragging his grand-child into it.
"The Name of the Doctor" takes that little speculation of mine a bit further, though. Spoiler:
So this "other incarnation" of the Doctor's is a complete mystery. The 11th Doctor knew who he was. The Great Intelligence implied that the Doctor's last incarnation was a brutal, evil, monster who led an army in violence and died in a final battle. The implication was that the "other incarnation" was that version of the Doctor, but the 11th knew him by face. Not really workable if the "other" was a future incarnation.
So what if this "other" incarnation is the original? What if the Time Lord known as the Doctor started out as... well, misguided. He got carried away and did terrible things. As a result, his family suffered and died. He regenerated and took a new name: "The Doctor". And thus the First Doctor was born. He goes and finds his only surviving family on Gallifrey, takes her, then steals the TARDIS. A legend is born.
That doesn't rule out the Doctor's story ending in a dark place, though. The 13th incarnation of this Time Lord (who would be the 12th Doctor) could still transform into the so-called monster who dies at Trenzalore.
I dunno. I have to say, I'm kind of digging the possibilities.