Formal:
* Emigrate: This is the most common and neutral term, meaning to leave one's country to live permanently in another.
* Relocate: This word emphasizes the physical move, but can be used for both temporary and permanent relocations.
* Expatriate: This refers specifically to someone living in a country other than their native country.
* Immigrate: This describes the process of moving *into* a country, but can be used to describe the act of leaving your home country to do so.
Informal:
* Move abroad: This is a simple and general term.
* Go overseas: This emphasizes the distance involved.
* Settle down in [new country]: This focuses on the intention to make a new home.
You could also use more descriptive phrases like:
* Seek a new life in [new country]
* Escape to [new country]
* Embrace a new culture in [new country]
The best word to use will depend on the specific situation and the tone you want to convey.