In ancient Greece how was the new settlement started by people who cut their ties with their native countries?

Answer 1

There were no 'countries' - there were independent city-states.

When these cities, with limited agricultural land, became overpopulated, they each shipped of their surplus to seize new land around the Mediterranean and Black Sea littorals, and establish their own city-states, resulting in the Greek world comprising over 2,000 independent cities.

These new cities maintained relations with their mother cities, sending them annual offerings, however these relationships often became strained resulting in later hostility.