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Immigration
There are three types of visas to enter to and stay in Mexico as a tourist or resident: 1. FMT (Tourist Visa) which is normally used by people that stay some months in the country and some other in their home country; 2. FM3 (Residence Visa), is the most used in the cases in general when people stay permanently in Mexico without losing the citizenship of their native country; and 3. FM2 (immigration Visa), which is very similar to the FM3.
To get these visas, depending on the one you need, it is necessary to go to the Foreign Affairs Ministry of the Mexican government (SRE) or the Embassy or Mexican consulate in you home country. There, they will tell you which documents you need to get your visa.
You could also do it to bring to Mexico with you your pets and belongings. Consult the regulation and documentation needed.