Was the language and traditional ways of learning suppressed? Did the catholic church take an active role in suppressing native beliefs and ways of life? I know that many of our sources on the Aztecs are spanish codices, that would seem to indicate that there are few native accounts to draw on. What's the story here? I assume that stuff was lost, but how much?
To get you quickly started, the overwhelming majority of Aztec codices did not survive the Spanish (or rival Aztec Emperors out to do some history re-writing) and were burned as a way of erasing their culture. So the answer to your "how much?" is "as far as we know, nearly all of it" until we make a chance discovery somewhere in the Central American jungle. This is also why almost everything we know about the Aztecs comes from Spanish sources. I can't speak knowledgeably about Aztec oral tradition.
It's misleading to focus on the destruction of codices as the key indicator of Aztec history. In no small part because this was only a minority of Aztecs were "literate." There was no fully developed writing system, but rather a series of conventions based upon widely understood iconography and rebuses. While the loss of Aztec codices is significant (Texcoco is said to have had an impressive collection), the fact that this was a particularly elite and particularly specialized form of recording history. The bulk of history was customary traditions and oral transmission.
Thus, a focus on codices ignores the works of Spanish friars and Nahua scribes that created an encyclopedic work of ethnography like Sahagun's General History of the Things of New Spain, a multi-volume compendium covering everything from dynastic genealogy to religious rituals to medical practices to bean selling. And this is merely the most significant work among many attempts by Spaniards to record the history and customs of the people they were living among, even as they attempted to turn them away from their culture and towards Christianization/Hispanicization.
We also have ethnohistorical works written by Nahuas after the conquest, often by descendants and relatives of the ruling nobility. Fernando Tezozomoc, Fernando Ixlilxochitl, and Chimalpahin all wrote invaluable works recording Aztec history from their own perspectives. Like the Spanish works, these have their own biases, but no more than any other historical primary text. Regardless, their status as recording the oral histories of the various peoples of Mesoamerica.
The assumption that Mesoamerican history ended in 1521, however, is a faulty premise. As I have written about here, the first couple decades after the "Conquest" basically saw business continue as usual. It would not be until major epidemics in the 1540s and then again in the 1570s; increasing confusion and dissension among the indigenous population over titles and claims; and the continuing influx of European colonists, that would see native culture precluded by Spanish. This was a process of decades and generations though, not something that happened in one day in August 1521.
Really though, the interplay between European and American cultures is one that continues to this day. The most significant events for today are not such much the distant actions of Conquistadores and Tlatoque, but the 19th Century movements to forge a common mestizo identity for Mexicans.
If you want a collection of Nahuatl primary writings, Lockhart's We People Here: Nahuatl Accounts of The Conquest of Mexico is a good resource. I'd recommend familiarizing yourself with the introductory texts on the Aztecs on the AskHistorians Book List first though, since the poetical stylings of Classic Nahuatl are not the most accessible. For a general history which specifically covers the building of a common mestizo identity, I'd recommend The Forging of the Cosmic Race: A Reinterpretation of Colonial Mexico.