Residency does not start in Mexico. You book an appointment at a Mexican consulate in your own country through the official appointment portal, attend in person with your documents, and if approved you receive a visa sticker in your passport. That sticker is not the residency card.
You then enter Mexico, tell the officer you are entering to exchange a residency visa, and within 30 calendar days present yourself at an INM office to complete the canje — the exchange for your actual residency card. Missing that 30-day window is the most common expensive mistake.