Shall and Should are both modal verbs.

1. Shall expresses certain laws and rules

Examples:
You shall abide by the law.
Students shall not enter this room.

2. Something take place or exist in the future

a) We use shall for offers

Examples:
Shall I pick you up from airport?
Shall we dance?

b) Suggestion

Examples:
Shall I open the window?
What time shall we meet?

c) Future expression and promises

Example:
I shall be here at 7 tomorrow.

Should is the past tense of shall

1. We can use it as a personal opinion

Examples:
You should go to the police.
You should see a doctor.

2. Use to express that we wish something had happened but it didn’t

Example:
You should have seen it, it was beautiful.

3. Should used to ask for someone’s opinion

Example:
What should we do now?

4. Used something expected or correct

Example:
Everybody should arrive by 6 pm.