Everyone had access to education, although then as now the nature and quality of said education depended on the social class you were born into. There was compulsory education in place, but not exactly a modern school format.

According to the Florentine Codex, the tlatoani Itzcoatl (1428-40) had all the historical codices burned, and this allowed the creation of a state-approved history (that omitted their subservience under the Tepanecs) that was then taught in the telpochcalli. So whilst you might call this universal education, an argument could be made that it was more like universal indoctrination.

The telpochcalli, or 'house of the young', taught commoner (macehualtin) adolescents about Aztec culture and history, as well as practical military studies for the men. A typical day would have included prayer, religious songs, weapons training, household chores and physical labour. Those who excelled at martial arts might be earmarked for a military career, but otherwise there was less social mobility than in the other educational institution...

The calmecac ('house of the lineage') was the more selective, theoretical school which primarily catered for the children of nobility (pipiltin), giving them the instruction required to become leaders, priests and scholars. There wasn't anything really resembling 'scientific' study short of astronomy, medicine, biological observation and natural philosophy. This was all combined with military education, albeit somewhat more geared towards leadership than that which was taught in the telpochcalli.

Although primarily for the children of nobles, according to the Codex Mendoza commoners sometimes could and did attend the calmecac (although the CM has a tendency to understate class difference in Aztec society). Unlike the telpochcalli, which was attended on a daily basis, students lived on site at the calmecac from the age of about six. Discipline in both institutions would have been fairly harsh by our standards - the Florentine Codex records various punishments for disobedient children ranging from beatings to piercing their skin with thorns, depending on their age.

To go back to your question, yes poorer people could and did get access to at least a basic form of education, which included social history, religion, and practical military studies. However, the children of nobility and exceptional commoners had access to a more select form of schooling that covered the above but also included reading, writing and proto-science.