In conversation, Mexicans often refer to their country as la república, short for la república mexicana, rather than using the word país. It’s a good habit to imitate.
Mexicans often mentally divide their country into three parts: el norte, el centro and el sur. These divisions are only partially based on north-south geography. Parts of the Yucatan peninsula lie north of Mexico City but the region is considered part of el sur. Perhaps Mexicans take into account that on a road trip starting at the northern border, the Yucatan would the last region encountered. Socioeconomics also plays are role here. El sur, almost by definition, is the poorest part of the country.
Guanajuato, Guanajuato State, Mexico
Here’s another common way to divide the country: Mexico City vs. everything else. Someone from Mexico City is formally called a capitalino or a defeño, but more commonly you’ll hear chilango. This latter term is often used deprecatively, but sometimes with endearment, especially by the chilangos themselves.
Mexicans outside of Mexico City often refer to Mexico City simply as Mexico. Voy a México. I’m going to Mexico City. You won’t find that usage within the city itself. The rest of Mexico called provincia. ¿Es de provincia? He isn’t from Mexico City? Provincianos tend to consider chilangos to be rude, stuck up or criminally inclined. In turn, chilangos tend to view provincianos as uncultured hicks. I’m just repeating the attitudes I’ve heard.
There are several non-obvious terms used to refer to people according to what part of Mexico they’re from. They’re worth learning. Tapatíos are from Guadalajara. Regio(montano)s are from Monterrey. Jarochos are from Veracruz, especially the port city. Poblanos are from Puebla. Poblanos may also be referred to by the offensive term pipope (pinche poblano pendejo).
Paisano/paisana (or paisa, for short) means fellow countryman, usually a fellow Mexican, especially one in the United States. A more general or formal term is compatriota.