What kind of education would a 12th/13th century English knight get?

They were not educated in the modern sense. They were not taught to read or write or philosophy. To become a Knight was a long and demanding task all by itself. It required years of training but that training was more practical. It was about horsemanship, archery, using a lance, a sword, a mace. How to take care of armor. How to walk, ride and fight in armor. Knighthoods were not bestowed. A young person, typically of nobel birth, would have started as a page, and then progressed as a squire. Young persons left their homes at early ages and entered into the service of great lords who could afford to train them. They would spend their lives in the service of their liege lord. So they had trained for years, even decades to learn what they needed to know to become knights and reading and writing were not among the requisite skills.

(I forgot about "English" in the title so this is more general.)

I'm reading Chris Wickham's Medieval Europe and just previous to your time frame one wasn't born anything; just a greater likelihood of receiving a title due to family and/or wealth, though it had to be earned and kept. Hereditary titles and a nobility were sometimes used especially in newly conquered areas; but any grant of land (in exchange for military service) was costly to the income of the king, count, lord, and so were given sparingly (widespread use of taxes was yet to come and that led to a hired standing army that changed everything). Military skill and managing farms would have been most important to education of knights. Holding the land was conditional on military service (many sources say 40 days a year was common) and if they couldn't fight well, they died. Administering their land well decided whether they would eat and be able to afford the trappings of knighthood.

Also important and left out of much popular history is that during this pre-Enlightenment era Greek language was frequently used in the deep administrative backbone of the Roman empire and in post-Roman leading societies - not Latin, thus suggesting education was Greek in nature. Not that education was largely conducted in Greek; Latin or local language were used; I'm referring to the trend setters.
So perhaps use Alexander's tutoring by Aristotle, Socratic dialogues, Ptolemy & Euclid, general classical Greek culture as examples. As far as I recall, T.H. White's Sword in the Stone is fairly consistent with this sort of pedagogy and even Game of Thrones follows the model.

Here we go; High Medieval Period in Wikipedia is worth a read and says Anselm of Canterbury had reformed/reintroduced education just like I said (mild surprise that my inference from Wickham's work was so accurate). The children were just looked after until age 7 (when according to Greek custom they were really people) then began some work and education. 12 yrs old they were introduced to Euclid as geometry and The Elements was considered the building blocks of all rational thought (children under that age mostly mimic, not grasp conceptual information).

Once again, fighting and farming being paramount of required skills. Interesting to note is that this period before the Black Death was quite prosperous and populations in Europe increased much. I would guess then that people skills were not to be ignored. Local knights (or whatever level of nobility administering an area) were also often required to serve as judges (depending on issue, along with local religious leader) and with rising population this would require more time and familiarity with Law.
(interesting is that Honor (and loyalty and all that) determined legal standing and so was often more important than wealth because without legal standing one couldn't defend one's self against allegations; if one person with standing said the other (without) had been paid for 40 sheep and not delivered, too bad for person without honor/legal standing even when untrue)

Of course religion was big, the Crusades during much of this period was conducted in Europe and was mostly inter-Christian slaughter. Basically if you wanted to eliminate competition or take someone's lands (perhaps to pay more knights via land grant) you paid your fee and applied for permission from Pope who would (after checking their counteroffers) declare them heretics for getting the number of angels dancing on a pin wrong and you get to slaughter and steal (liberate). So some religious training would be important, not necessarily much actual education as training in how not to get into trouble with the Church (whoever was eventually going to win though not so sure at the time which version would come out on top) was pertinent. I don't know if knights had much power here as they were sworn to whichever lord and rose or fell with him. Papal bulls promising get-oughta-hell-free cards to knights participating in Crusades was often meant to allow knights to abandon their sworn lord without losing honor and standing (during European Crusades era). Most likely these knights were never trusted again and eventually had to be shipped off to war in Muslim lands no matter what the Pope had said (I tracked the ebb and flow of inter-Christian crusading with external Muslim crusading and this did seem to be the pattern). However "holy" the motive, disloyalty was akin to "betray your family" accusation that is difficult to wash off no matter the circumstances. Perhaps some formal religious education would have helped if dealing with a powerful ambitious local clergy. (I have referred to European Crusades without intending to include Spain as during the OP's timeframe this was a stalemate).

Knights needed education/training in Fighting, Farming, Law, People Management, and luck in religion.

Update: Adelard of Bath worked out a whole Liberal Arts curriculum (grammar, rhetoric, logic, mathematics, geometry, music, and astronomy) around 1120 CE though it appears mostly kept to scholars until 13th century. This does allow for the possibility of a highly educated knight if he had access to Bath Cathedral (115 miles west of London in southern England near Bristol). Adelard wanted to educate everyone but printing wasn't available yet and books quite expensive. (thanks Pieter Geerkens for asking right questions)