If you plan to reside permanently in Mexico, you should take the Mexican nationality and renounce to your original one. The procedure takes between 6 to 8 months.
According to the Political Constitution of the United Mexican States, there are different ways to achieve the Mexican nationality, the first is by residence, all foreigners living in Mexico for over five years, can automatically make the procedure to establish the Aztec nationality by having the FM2 and FM3 immigration forms issued by the Interior Ministry.
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There are some cases where it may be approved earlier, for example to have children in Mexico or marry a Mexican man or woman, only requires two years of residence. Similarly as if you were born in a Latin American country or the Iberian Peninsula or if you are a direct descendant of a Mexican by birth.
If you were born abroad but one of your parents is Mexican and so as their parents, you are considered Mexican by birth, so the only thing to do is that your parents attend the nearest Mexican Consulate to their birth place and register you, or register the foreign birth certificate (translated into Spanish) in the Mexican civil registry to get you the dual citizenship.
As for the military papers in the case of men, if you have dual nationality, you’ll be free from this service; you would only have to go to the Secretary of National Defense to the delivery of your document.
The Naturalization letter is a legal document by which a foreigner acquires Mexican nationality and also Mexicans rights and obligations. However, you can never hold public elected office.
As said, if you get the Mexican citizenship by naturalization you must give up your last nationality, and to be able to leave the country you must apply for a VISA and a Passport as any other Mexican.
Remember that if you cross the border into Mexico by car just for a few days, you must arrange your temporary import permit as is required, in addition, you should get a Mexican auto insurance, in case of any problem your car would be covered with an insurance policy.
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Source: sre.gob.mx