Use persons in:
Legal or official documents (e.g. police reports, formal notices)
When emphasizing individuals separately, especially in technical or bureaucratic language

Examples:
Missing persons report
Talking about groups of humans in general
Any person or persons found trespassing will be prosecuted

Use people in all other contexts.
Everyday conversation
Talking about groups of humans in general
Describing communities, populations, or social groups

Examples:
There were five people at the meeting.
People are getting tired of waiting.
Young people today face different challenges.

Peoples is different. Peoples is used to talk about different cultures or ethnicities. 

Examples:
The indigenous peoples of North America have diverse traditions.
War had displaced many peoples across Eastern Europe.
The museum has an exhibit on the art of African peoples.
Anthropologists study the languages of ancient peoples.

Note:
Shall might still be used in British English, but it’s not used in American English. You may hear us use it facetiously or ironically – we are being funny. And you may find it in legal language or older literature.