Shall is traditionally used with I and we to talk about the future or make suggestions.
- Shall we begin?
- Shall I open the window?
It is particularly common in British English for suggestions and offers.
Should is mainly used for advice, expectation or obligation.
- You should get some rest.
- We should arrive by six.
- You should check the details.
Easy guide
Shall = suggestion or offer.
Should = advice or expectation.