If you wish to stay for more than 6 months consecutively in Mexico then you will need a Resident Card. The card is issued for the first time outside Mexico. So first time applicants need to go to their nearest Mexican Consulate outside Mexico.
It is good for 4 years after which you must either leave the country to apply for a new card or convert it to a Permanent Resident Visa (MPRV) / Residente Permanente.
If you already have non-immigrant status (FM3) in its 1st or 2nd year then you can convert it to a temporary resident visa at the time of renewal of your status without leaving the country.
If your non-immigrant status (FM3) is in its 3rd or 4th year then you will have to convert it to a permanent resident visa or leave the country. You can apply for a change in status 6 month prior to its expiry.
The MPRV can be purchased for up to 4 years without having to renew it annually.
There is a minimum income requirement for getting this visa in the range of $2,000 USD per month or $102,000 USD of investments for each individual. You may be required to prove this with bank statements but the Immigration offices have the discretion to use a combination of assets including real estate to arrive at an equivalency. The idea is that you should be financially self-sufficient.
Once you have met with the Consulate they will provide you with a VISA document inside your passport. With this document you can legally enter Mexico and you have 30 days within which to present it to the local immigration office to exchange it for a valid Temporary or Permanent Resident Card.
Once you have the Resident Card, on it will automatically appear your new CURP.