Is it safe for an American to move to Mexico? Where is all the fighting?

Yes it is. Most violence is in border cities. There are over ONE MILLION Americans living in Mexico and we are just fine. You can live here for years and see no violence. Most people live their lives here untouched by cartel activity or violence. There are many great places to live in Mexico. Your primary issue is getting a residence visa...it is not easy. You can forget getting a work visa. if you do not have a steady income from the U.S., like a pension...it is nearly impossible for you to live in Mexico. See site below for great info and the visa info. you will need an appointment at the nearest Mexican consulate to begin the residence visa application BEFORE coming to Mexico. They can explain the process. Remember, bureaucracy is an art form in Mexico. If it is possible to complicate a process, they do. Of course, getting a residence visa is not easy in most countries.

People from the United States and from other countries just do not up and move to Mexico.

I will need to qualify for permanent resident, i.e. about 2,000 dollars per month income.

You cannot work, unless you get a work visa, and unless you are a nuclear scientist or a petroleum engineer you probably will not get one, because the employer must certify that there are no Mexicans that are qualified for that job. If you quit, your work visa is not valid, and you would have to find another employers that jumps through the hoops to get you another one.

As far as safe, where are you talking about. Mexico is a whole big wide country and only a very small percent of the area is not safer than most places in the United States.

A little racist btw, and it's completely fine if you know fluent Spanish and the fighting a just like any other place. It all depends where your going