- There is not really any direct answer to your question.
- The Emperor does not teach anyone, nor the Imperium as a whole.
- It is doubtful that a great many are literate.
- Through stories or necessity there is knowledge of other worlds, important worlds, the types of Xenos, and the victories of the Emperor.
- Public education depends entirely upon the world, status, and expectations of each individual.
- In general, there is a desire to keep the public uneducated.
So there is not really any direct answer to your question available. I am answering this based upon what I know, but you will really have to extrapolate and guess beyond that yourself. It is best to imagine the Imperium on the scale of a medieval population. Kings, knights, soldiers, commoners, outcasts, etc...
Key to your question, there is no Imperial wide school system. No one grows up learning the about the golden age of technology or the treachery of Horus in school with kids in class doing reports and standardized testing comparing the intelligence of each world.
That said however the Ecclesiarchy maintains the Schola Progenium. This school system is filled with orphans of all types and teaches the Imperial Cult. Graduates of this academia often become Commissars, Sisters of Battle, Imperial Guard officers, etc.
There are also a number of other scholas that educate based upon their own doctrines.
In addition, there are a number of High Born societies that exist in the universe that would almost guarantee their children some sort of schooling and greater intelligence than the average commoner (think of hive worlds in particular where the elite class lives near the pinnacle).
Because most individuals are born into a class based society literacy is not always important. If you will just be working in a promethium mine your entire life you have no need to learn to read, whereas if you are expected to become a Guard officer or planetary governor like your father you will likely learn to read and have a broader educational experience than the miner would out of necessity.
Generally speaking, most knowledge is passed down from father to son or learned through the Ecclesiarchy and most become embellished or myth. Only the details that perpetrate the desired effect are passed on and they are changed to better suit the needs of those that are in control, whether as wide as the Imperium or as a mother simply telling her child the Eldar take kids that don't eat their vegetables. It is easier to control the ignorant, especially if you can control what they do learn.
This dynamic applies to both your questions of the Emperor's victories and the knowledge of Xenos. While some learn the "true" history, most learn what they know from hearsay or the statues and stained glass long since erected. An example of this can be found within the first issue of the Damnation Crusade comics...
A great quote that also applies is from Bjorn the Fell-Handed (a Space Wolf that walked with the Emperor prior to the Horus Heresy) where he said "God-Emperor? Calling him a god is why this mess started in the first place."
Another good example of this is the difference of High and Low Gothic, the primary languages of the Imperium. Low Gothic is the common tongue and High Gothic is reserved for privileged individuals and sometimes even considered holy.
Most specifically, some worlds do not even have the opportunity to learn those more common things however. For example, the citizens of Fenris have no statues or Ecclesiarchy to consult and they do not even speak Low Gothic. They grow up learning what is called hearth-cant and battle-cant. The Space Wolves routinely speak this themselves when they do not want outsiders to know what they are discussing, but thanks to the knowledge machines they also know both High and Low Gothic for when they need it.
There are a number of quotes that exist in the universe, typically said by Inquisitorial members that perpetrate the idea of under-education. I am sure there are more, but I remember these off the top of my head.
"Knowledge is power, guard it well."
"An open mind is like a fortress with its gates unbarred and unguarded."
"To question is to doubt."
"A small mind is easily filled."
"A good soldier obeys without question."