I am an immigrant.

You are liable for US income taxes on your global income from all sources as long as you are a citizen. If you receive non-US source income, you must declare it on form 1040 each year. There is a foreign earned income exemption form you can file, that will exempt the first $80K of foreign earned income from US taxation. That will avoid double taxation on your earned income. If you have: a) any amount of non-wage income from foreign sources or b) earned income from foreign sources in excess of $80K on which you are taxed by a foreign entity, you can reduce both types of income tax by using foreign tax credits to offset the US tax. If you own property in a state, you will be subject to state property taxes and you can offset this from your foreign tax in some cases, depending on the tax laws of the country.

Becoming a permanent resident of a foreign country will not affect your US citisenship. Becoming a citisen of foreign country will only lead to termination of your US citisenship if: a) the foreign country does not allow double-citisenship, or b) for some strange reason other than a) you appear before a US consular official and renounce your US citisenship. That is to say, the US allows double citisenship, but some foreign countries do not. By the way, if you renounce your US citisenship you will lose any benefits you may have accrued through social security or US military service. On the other hand, you can receive US benefits as a foreign resident provided you are owed some. Some countries – mainly in Europe – have social security treaties with the US that influences your taxation and accrual of US benefits. I am not resident in such a country. I pay no US social security on my danish income and recieve no benefits from that, either. However, on my retirement I will still be eligible for the benefits I accrued before I immigrated.

You can not legally be a resident of the US and another country (US Law), but you can be dually domiciled. But I wouldn’t bother doing that, even if you own property in the US. Being a resident of another country while holding US citisenship will in no way affect your right to enter the US freely, own property in the US, or collect income from US sources. (And you will be taxed on it anyway). In fact, if you maintain your US residence, you might have to pay state income taxes in your resident state. So, just be a resident of where you really are. It’s simpler and you don’t lose anything.

Those are the basics.

I know nothing about Peru, except I went to prep school with the grandson of a former ousted Peruvian dictator.

Bon voyage, if you make it.