I have got my passport, but if I want to go abroad, do I need to go through some local formalities, such as health certificate?

After getting a passport, you have to apply for a visa to the destination country. You have to apply for this at the embassy or consulate in China, the destination country, and you have to apply for an international travel health certificate. You can go abroad after going through these procedures at the international travel physical examination center of the city (or above) epidemic prevention station.