Should we fast according to the calendar followed by Muslims in a non-Islamic country, or should we fast according to the calendar of our home country?

The crescent moon is established by legitimate sighting with the naked eye, not by astronomical calendar. Therefore, a Muslim should seek to sight the crescent moon himself. If he sights the Ramadan crescent, he fasts, and if he sights the Shawwal crescent, he breaks his fast. He may also rely on the sighting of nearby countries whose moonrises do not differ from his own, and there is no harm in following those who do not consider the differences in moonrises.